Hi. What you got so far seems right.
Now i can't think of any easy "trick" to show the uniqueness of E in this representation, so i would suggest you work on how to enforce:
<φE(p)|φE'(p)> = δ(E–E')
I suppose you can look up the integral; given the result you should see:
1 - how E needs to be positive
2 - how E must be = E' (i.e.: E is unique)
That's a start...