You're in luck,
The Road Goes Ever On, by Tolkien and Donald Swann is just what you're looking for. It contains music for Tolkien's songs written by Swann and approved by JRR Tolkien, along with a number of other interesting things. The new edition (now with a CD) has just become available in the UK for the first time in 25 years (thanks, no doubt, to the movies). I'm not sure about avaiability in the US, or other places, it may be more difficult to obtain there.