 formal styles, while accusative version ("us") would be considered to have an informal flavor.
Here's a related excerpt from the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language, section "Constructions where nominative and accusative are in alternation", pages 459-61:

There are a number of constructions where both cases are found. In most, the nominative is restricted to formal (or very formal) style, with the accusative appearing elsewhere.
(a) Subjective predicative complement
[9]

*

*i. a. It is I who love you. - - - - b. It's me who loves you.


*ii. a. It is I she loves. - - - - b. It's me she loves.
...
Probably the most frequent use of a nominative case predicative is in the it-cleft construction, as in [i-ii]. And here we can make a distinction according to whether the pronoun would be in nominative or accusative case in the non-cleft counterpart: compare [i] with I love you and [ii] with She loves me. In the former, the accusative version [i.b] certainly has an informal flavour, whereas in the latter the nominative version [ii.a] seems very formal and accusative [ii.b] relatively neutral in style.

Notice that in the above excerpt, it is example [9.i] that is related to the OP's example.
Now as to the supplement "Kim and I/me", I'd think that for many speakers that it would probably be preferable that the supplement's pronoun case be consistent with the pronoun case of its anchor (the anchor is the predicative complement in the it-cleft's main clause). But this topic is basically unclear to me, as accusatives are often the only option or the preferred option for many types of constructions (e.g. "Me, I wouldn't trust him further than I could throw him"; "Well, there's always me, I suppose"; "Yes, there's certainly her to consider" -- page 462, CGEL).
And so, these are the two versions that I've been looking at:

*

*1.) Our supervisor finally noticed that it was we, Kim and I, who always turn in our reports on time.


*2.) Our supervisor finally noticed that it was us, Kim and me, who always turn in our reports on time.
Both versions are grammatical and acceptable in today's standard English; the difference is a matter of style.

A: My gut says this sentence should read:

Our Supervisor finally noticed that it was us, Kim and myself, who always turn in our reports on time.


A: It would only be "you and I" if you were directing your speech to Kim. "Our" works with the third person grouping "Kim and I" because you're telling the third party audience that the supervisor belongs to you and Kim. However, I do think your sentence would be better if you left out "we" altogether. It's redundant, and not in a clarifying manner. 

A: Well, if the sentence isn't being addressed to Kim, Kim is right.
If it is being addressed to Kim, then it would normally be you, though referring to the second person as if they were a third person is sometimes done.
I is correct.
I'd definitely go for "it was... who always turned in" or "it is... who always turn in" though.

A: This is debatable. One argument is that "it was us/me" is wrong because "us/me" are the subject and should thus be "it was we/I". However, common usage is often "it was us/me". Hence your options are:
"Our Supervisor finally noticed that it was we, Kim and I, who always turn in our reports on time." 
"Our Supervisor finally noticed that it was us, Kim and me, who always turn in our reports on time." 
If speaking to Kim:
"Our Supervisor finally noticed that it was we, you and I, who always turn in our reports on time." 
"Our Supervisor finally noticed that it was us, you and me, who always turn in our reports on time." 
Seasons, Readings, Writings, and Thanksgivings
I've been a bit MIA around these parts lately. Part of that is due to our month long fast and the other things my fingers found to keep busy. Part of that is just that it's winter and winter, for me, has always been a hibernation time. I think God created the seasons for a reason and he means for us to live into them instead of living into the seasons we make for ourselves. I think part of the reason our world is so tired and hurried and anxious is because we are constantly trying to force unnatural rhythms onto life. We take vacations in the summer and fill our autumns and winters with activities galore, never minding that God designed summer for growth, autumn for harvest, winter for rest, and spring for planting. If we were to truly live into those seasons just as they are, I think we would be less prone to throw around words like "contentment" or "season of life" or "exhausted" as lazily as we do. God meant for winter to slow us down, to slow our production, to sometimes cease our growth, and to let dead things die if they must. And none of that is bad. It's just our perspective that needs to change.
Also, though, I've been sick the past week and it's easy to talk about hibernating when you can't breathe out of your nose or your mouth and when your head feels like it's under twenty feet of water. So there's that. But also, seasons.
I read a lot throughout January and although most of my reading wasn't online, I did read a few pieces I wanted to share with you. They might interest you too:
I cut this one out of our Sunday Times and taped it to our fridge I loved it so much. The Poet of Light by Christian Wiman on Richard Wilbur.
If you've seen Look & See: a portrait of Wendell Berry, then you probably had the same complaint I did: we hardly saw anything of Berry himself! But something I loved about the documentary was the delightful presence of his wife. Here's an article on her that made me want to be a wife like she is.
This was a quiet podcast for a quiet evening, both of which I quite enjoyed. Krista Tippett interviewed John O'Donohue for OnBeing.
Speakings of podcasts, Nate and I worked through this series from Beau Hughes (The Village Church, Denton) on shame. I cannot recommend it more highly.
I hope you took a few minutes to read Rachel Denhollander's words at the conclusion of the Larry Nassar trial. This is a great follow-up interview at Christianity Today with her.
I subscribe to Poetry Foundation's Poem of the Day feed and you might want to as well. Listening to poetry is such a good discipline. Reading it is fine and good too, of course, but poetry is lyrical and best experienced heard.
Also, I just wanted to say a heartfelt thank you to all of you who support Sayable on Patreon and who have downloaded the e-books. I have gotten so many messages from you saying you're being encouraged by the work there. That means so much to me. Thank you. Thank you. Thank you.
I am only 30 away from 200 supporters on Patreon, at which point I'll be starting to coach a small writing group. Details about it will come after that point, but I will say there will be an application process and it will be opened first to Patreon supporters. I will only be able to invite 20 people into the group (which will last between 12-16 weeks, still undecided on that), so if you are at all interested you'll need to begin preparing a 300 word non-fiction writing sample (on anything). There will be a cost for participation in the group, but it won't be astronomical, just to cover my time coaching. I read through my tentative plan to Nate last week and started getting pretty excited about this endeavor. Everything we'll be doing has been part of my process of becoming a better writer, thinker, and receiver of criticism. I hope it helps each of you as well. Again, more details on that after we reach 200. Grateful for each of you.
View from the sickbay. Harper is under there somewhere...and Nate, I think.There are a number of reasons lawns can become unhealthy. Sometimes harsh summer heat or winter frost can cause your lawn to deteriorate. Other times, it might just be lacking a little TLC. The good news is that there’s a few really...
Can you believe that summer is just around the corner?? While you might be hanging out for the hotter temperatures and longer days, your lawn definitely is not. Summer is undoubtedly the toughest season for lawns in Australia and we need to...
Hallelujah, winter is finally over and we’re well on our way to prime growing season! To help get your lawn looking bang on, check out our top 5 lawn care products you should definitely have in your shed for spring. Maintain 26-2-9...
While it’s fantastic that winter is finally over (both for our lawns and for us), now is a time