ut I'm also a beginner in Polish so may well be mistaken
Don't worry, Michal is a beginner too.
I thought mi przykro was used in sense like 'I'm sorry that your friend died; I'm sorry you are not feeling well. I thought mi przykro has more of a sense of 'being upset' rather than the sense of 'an apology'.
Przepraszam - for something you had an influence on. (you chose the moment to reply to the letter, so it's your fault, you might have been really busy etc., but you set your priorities, and replying to someone's letter wasn't at the top of the list)
Przykro mi - for something beyond your influence.
"Przykro mi, ale to co mówisz jest bez sensu" - "I'm sorry, but what you're saying makes no sense." - Here you're making an unpleasant comment, so to attenuate it a little you can add "przykro mi", but you don't have to apologise for it, because it's the other person's fault that his/her words make no sense.