Hint: you need to use the sum rule.

The computation is quite manageable for the case of k=5. For the general case, I too was left feeling dissatisfied with the expression I found, but on reflection I'm somewhat confident it is the correct answer.

The case k=4, Ni=13, m=5 is solved numerically on a Web site which discusses probability for Poker players, that was helpful in checking my results; the answer to 3.2 is a generalization of the results given there.

There does not appear to be a complete collection of solutions. This site comes closest. If I were you I would avoid looking at their solution for exercise 4.1 (I'm trying to forget what little I've seen of it as I'd like to solve 4.1 under my own power), but I would also not feel bad about giving up on 4.1 if you find it difficult.

I'd be happy to discuss Jaynes further over DMs or email - though I may respond at a slow pace, as I'm working through the book as my other activities allow. I'm on chapter 6 now.