If talking about the past - I did all I could. If talking about the past leading continuously up to and including the present - I have done all I can. This suggests that there may be more that could be done (by someone else) about a situation, but you can do no more.

Yes, since you are talking about the past. "I did all I could" makes more sense.

@A*P Thank you. I think I get the sense of it...

I did all I can, is correct but makes no great sense -- it refers to the skills I *always* have: but what's the point to pretend such skills that proved terribly insufficient... are we kidding? That's why "I did all that I could" is almost an idiomatic expression -- because there's simply no other way to say it.