Here was the set of tiles that I had at the end:
I decided to save the tiles and try to work out a solution later. I spent some time working on this problem on two consecutive nights. The second night I found a solution that used all the letters but one N, but that seemed to be the best that was possible.
Finally, several days later, I found a true solution. It's possible to vary some of the peripheral words and get alternate solutions, but there is a core structure that I have not been able to alter without rendering the grid uncompletable.
This puzzle can be solved without using any two-letter words or any vowelless words. (I think that violating these constraints would make the puzzle too easy.)
I leave this as a challenge. I will post a solution at some point in the future.