What people are calling "waxed paper" above is often called "greaseproof paper" in the UK, in cae that helps.

You say you have trouble getting the 'rectangular' end of the kazoo in your mouth. Here's an idea Mo - try playing the kazoo the other way round, i.e. insert the narrow end in your mouth. It should still work, as you are blowing down the tube and should be able to cause the paper membrane to vibrate. Worth a try, and if it works, why not? (ISTR when I was a kid and was given a kazoo, never having seen one before, I played it the 'wrong' way round, as I assumed the wide bit was away from your mouth, as with most wind/brass instruments! It still worked OK though, as I recall).