I do not give you a reference for the existence of connections on any principal fiber bundle, but just of linear connections.
Sure your question was about connections over principal bundles and, on page 67 in KN, there is a complete proof, but, on page 68, they just remark the possibility of another proof.
This alternative possible proof is the one you do not have understood.

So in order to help you to overcome this difficulty, I refer you to Lee's textbook on Riemannian Manifolds.
Even if his Lemma 4.4 and Proposition 4.5 give all the details of the proof only for linear connections, once you understand what you was missing, then the same argument works in general.

I hope to have been useful.