How do I write a letter that is both contrite and yet also asks for something?

About a year ago, a former boss wrote me an email telling me that a child in the family was very sick. I was well-intentioned, and planned to send flowers and a card. In the end, I did nothing (not even respond). Time passes. Much guilt accumulates.

Today, I need a letter of recommendation from this former boss to apply for another job. I really want to maintain a good relationship with this person, but also absolutely need this recommendation to be sent within a month.

So, how do I write such a complex letter? Should I break it up into two letters separated in time to diffuse the nakedness of the motive? Or is there a way to say “Sorry I didn’t respond when you wrote to me a while back. I hope everything is going well. By the way, I need a recommendation….”?

If it makes any difference, I’ve found out through channels that the child is doing better now.