<s>[INST] Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$ Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$
where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number.  The $A(p,q)$'s are known as alternating Euler sums.

Can someone provide a nice proof that 
  $$A(1,1) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2?$$

I worked for a while on this today but was unsuccessful.  Summation by parts, swapping the order of summation, and approximating $H_k$ by $\log k$ were my best ideas, but I could not get any of them to work.  (Perhaps someone else can?)  I would like a nice proof in order to complete my answer here.
Bonus points for proving $A(1,2) = \frac{5}{8} \zeta(3)$ and $A(2,1) =  \zeta(3) - \frac{1}{2}\zeta(2) \log 2$, as those are the other two alternating Euler sums needed to complete my answer.

Added: I'm going to change the accepted answer to robjohn's $A(1,1)$ calculation as a proxy for the three answers he gave here.  Notwithstanding the other great answers (especially the currently most-upvoted one, the one I first accepted), robjohn's approach is the one I was originally trying.  I am pleased to see that it can be used to do the $A(1,1)$, $A(1,2)$, and $A(2,1)$ derivations.
 [/INST] Interestingly, $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}H_{n}^{-}}{n} = \frac{\zeta(2)}{2} {\color{red}{+}} \frac{\log^{2} (2)}{2}$$ where $H_{n}^{-}$ are the alternating harmonic numbers defined as $$H_{n}^{-} = \sum_{k=1}^{n} \frac{(-1)^{k-1}}{k} .$$
One way to show this is to notice that $$ \begin{align} \log (2) - H_{n}^{-}  &= \sum_{k=n+1}^{\infty} \frac{(-1)^{k-1}}{k} \\ &= (-1)^{n}\sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k+n} \\ &= (-1)^{n} \sum_{k=1}^{\infty} (-1)^{k-1} \int_{0}^{1} x^{k+n-1} \ dx \\ &= (-1)^{n} \int_{0}^{1} x^{n}\sum_{k=1}^{\infty}(-1)^{k-1} x^{k-1} \ dx \\ &= (-1)^{n} \int_{0}^{1} \frac{x^{n}}{1+x} \ dx . \end{align}$$
Thus an integral representation of the alternating harmonic numbers is $$ H_{n}^{-} = \log (2) + (-1)^{n-1} \int_{0}^{1} \frac{x^{n}}{1+x} \ dx .$$
The integral on the right can be evaluated in terms of the digamma function, and you'll get a closed-form expression for the alternating harmonic numbers.
But getting back to evaluating that sum, 
$$ \begin{align} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}H_{n}^{-}}{n} &= \log(2) \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} + \sum_{n=1}^{\infty} \frac{1}{n} \int_{0}^{1} \frac{x^{n}}{1+x} \ dx \\ &= \log^{2} (2) + \int_{0}^{1} \frac{1}{1+x} \sum_{n=1}^{\infty} \frac{x^{n}}{n} \ dx \\ &= \log^{2} (2) - \int_{0}^{1} \frac{\log (1-x)}{1+x} \ dx \\  &=\log^{2} 2 - \int_{1/2}^{1} \frac{\log \big(1-(2t-1) \big)}{2t} \ 2 \ dt \\ &= \log^{2}(2) - \int_{1/2}^{1} \frac{\log \big(2(1-t) \big)}{t} \ dt \\ &= \log^{2}(2) - \int_{1/2}^{1} \frac{\log 2}{t} \ dt - \int_{1/2}^{1} \frac{\log (1-t)}{t} \ dt \\ &= \log^{2}(2) - \log^{2}(2) + \text{Li}_{2}(1) - \text{Li}_{2} \left( \frac{1}{2}\right) \\ &=  \zeta(2) - \frac{\zeta(2)}{2} + \frac{\log^{2} (2)}{2} \\ &= \frac{\zeta (2)}{2} + \frac{\log^{2} (2)}{2} . \end{align}$$
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