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The 44-year-old, who won five league titles and the Champions League with United, contracted an airborne virus.
"I don't think my kidney is going to be 100% now, if it gets back to 70-80% I will be delighted," he told BBC Sport.
"When I admitted to myself I was ill, it was tough to accept. I didn't want my family to worry about me."
Cole, who scored 121 goals in 275 appearances for the Red Devils, played for 12 clubs in his career - including Newcastle United and Blackburn Rovers - and made 15 appearances for England.
He has been diagnosed with a condition called Focal Segmental Glomerulosclerosis, which causes scarring of the kidney.
"Putting on all the weight has been the toughest part," added Cole. "When I retired, I promised my kids I will never put on weight because when [footballers] retire they eat and drink.
"Seeing the change in my body, that has been tough, it wreaks havoc with yourself. You get comments like, 'Andy Cole's let himself go, who ate all the pies?'
"I have to take it day by day. I am four months off a year and it has been tough going through this process."
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Former Manchester United and England striker Andy Cole wants to "go back to things like going for a run and playing football" after kidney failure.