I doubt closing the cards with late payments will delete them from your report. If that were the case people would be doing that. I think your best bet is time -- just keep making on-time payments.
Closing wont remove the derogs from your credit report/scoring.
You could either request good-will deletion of at least the 90-late, or you could request the creditor delete the entire account.
Thirty and 60-lates are not major derogs. It would be nice if they would delete all derogs, but getting the 90-late deleted would most remove the major derog.
The account thst was only 30-late is not a problem, as that is only a minor derog.
Search these forums for good will...You also should visit and post in the rebuilding forum as there is a wealth of information there.
If you do not get the 60 and 90 day lates deleted, they will impact your score for the duration of 7 years.
My question is, is it safe to close my capital one and credit one that has the missed payments? Will that improve my payment history and remove the "serious delinquency" from my credit?
Nope -- closing does not remove negative information. The impact of lates and derogs is why you must avoid incurring them in the first place. These are serious red flags to creditors as they indicate that you have or had issues with paying your debts. Defiintely do whatever it take to avoid them. As for attempting to remove them -- hit the Rebuilding subforum and carefully research to see what you can do.