The STI integrated shifters you are referring to are for a 2x10 setup. There is no STI that I have ever seen that will work with a 2x5.
Minimally you are looking at two STI shift levers, cables, housings, crankset, cassette, wheelset, chain, front and rear derailleurs...
I did this conversion on my 1989 Paramount, but it was more cost effective for me since I worked at a bike shop which had bins upon bins of used parts, so all total I had about $150-200 in used parts. You couldn't buy the STI shifters for that alone I'm afraid, at retail prices.
Truthfully if you are restoring it, do just that, don't bother upgrading, and enjoy it for what it is. You will spend as much as a new road bike costs if you choose to upgrade, figuring you can get an entry level roadbike around $600. Especially if you can't do all the work yourself.