How do I relieve horrible pain almost 2 months later after crown?

While it is possible that the roots are short and in the "big picture" removal is a longer lasting option (with a dental implant), root canal therapy may be a good choice.  There are three options:  do nothing, remove the nerve, remove the tooth.  If the final option is taken, there should be a strategy on what to do to replace the missing tooth, as removing it will be a chain reaction, domino effect to the bite and adjacent teeth.

If a new crown is "high" in other words doesn't mesh properly with the opposing tooth, the pain can be very uncormfortable, especially a month after placement. Did your dentist adjust the occlusion? That is the first step.If no relief then a root canal is needed. That will probably resolve the pain, if not even after a root canal, then an extraction is necessary. If severe pain persists after  a root canal then the tooth is probably fractured down the length of the root. Don't extract until you have done the above steps. The length of the root is irrelevant…..

The question to be determined is whether or not your pain is "reversible", and will ultimately go away, or irreversable, and won't go away or will "finally" go away when the tooth dies.Ibuprofen is good for post dental sensitivity, but you should not have to have to put up using it for a long time.I think you might have misinterpreted what your dentist was trying to say.  Short roots do not have much significance into causing tooth pain.In general, all other things being good, I agree that you should consider a root canal before removing (extracting) your tooth.If your dentist does not do root canals, I would STRONGLY suggest that you seek a second opinion from an endodontist (root canal specialist).  The endodontist would be best qualified to determine if the tooth needs a root canal to relieve you of your pain, also doing root canals is their specialty.  One other thing that leads me to think you would be a candidate for a root canal is that it has been "two months" that you describe the pain as being "horrible."  In reversible situations, the pain generally lessens over time, and it would seem that from your description that is not happening.See an endodontist!