How can I pin murder on Eli, the pedophile?

The short answer is that there is none. The lack of hard evidence against both suspects is supposed to nag at you. The captain wants you to accuse the pedophile, as he's the better one to put behind bars.

I can't reveal what is going on, though, without massive spoilers:

Neither man murdered Diedre - they're both innocent. In fact, most of the murders you investigate while on the Homicide desk are actually being committed by Garrett Mason, a truth you'll eventually discover in the final case of the desk. The flimsy evidence you have against the two men is actually part of Mason's plot.