Budiansky is a good writer: his "Battle of Wits" about WW2 Enigma cryptanalysis is worth reading, even though it is mostly rehashed from other sources. I'll probably try to check out this Gödel biography, though the mental illness angle sounds pretty bogus.

But it seems to me, we already have a perfectly good Gödel biography, "Logical Dreams" by John Dawson, who is himself a logician, catalogued Gödel's Nachlass which probably made him the world's foremost Gödel archivist, and was also one of the editors of Gödel's Collected Works (5 vols).

Obviously I haven't read the new book so I can't compare directly, but my Bayesian prior is that if you're going to read just one, go with Dawson's.

If you're more interested in understanding the incompleteness theorem and the great amount of nonsense and confusion surrounding it, try "Gödel's Theorem: An Incomplete Guide to Its Use and Abuse", by Torkel Franzén, reviewed here:

http://www.ams.org/notices/200703/rev-raatikainen.pdf

Dawson's biography of Gödel is reviewed here:

https://www.ams.org/notices/200108/rev-davis.pdf