Okay, what about this ... ?
If you can, get the app downloaded to your iPad (as you've said it has installed before).
Then, hook your iPad up to iTunes and, with a right click on the iPad line, select "Transfer Purchases." This should put that app into iTunes.
Then, on the right hand side of the iPad/iTunes view, go to the Apps tab . Scroll through the apps until you find the one you've been messing with; it should be checked. Uncheck it.
Then, you should see an "apply" button at the bottom right side. Click that and iTunes should sync that app off of the iPad. Once done, take a look at your iPad screen. Hopefully, that app should be gone. If it is, eject the iPad.
Then, again in the left side, go to the top, under "Library," and select Apps. Scroll through the list of apps in your iTunes until you find that particular app. Right click on it and select Delete (and, choose to move it to the Recycle bin as you don't want it saved anywhere).
Now, this all is a stab in the dark, cause I honestly have NO idea why you're having these troubles. But, this is what I'd do given your situation.
Good luck and let me know how it turns out.
Marilyn