Saying why might be overkill, and isn't necessary to the apoplogy. The point is you are grateful for the gift, and you acknowledge that the thank-you should have been sent sooner. To divert into illness, etc. may be (incorrectly, but understandably) construed as an attempt to excuse the behavior and get sympathy. I think an apology that includes an "excuse"* is usually unsuccessful and comes across as insincere.
*I understand that sometimes apologies need explanations, but I'm talking about "I'm sorry I slapped you in the face this morning, but the coffee in the lounge was cold and I accidentally put salt on my cereal..." which really isn't apologizing at all, IMHO.