I Don't know how I can I make my kitchen sink drain down pipe intersect with the "p" trap?
Sounds like you need to install a new P-trap. Cut it off past the trap and then re-plumb in the correct spot. You may be able to use the old P-trap depending on where you need to cut, how it's glued, etc, but without a picture that's impossible to judge.

Be sure when you cut it that you'll be able to get a new fitting on the end, that usually means at least 3/4" pipe sticking out into the cabinet space.

Dry fit the entire assembly first, including the tailpiece to the sink, and only glue it once it's all in place. 

There are P-traps with unions on one end that can help if the fit is tight, and regular unions (or slipnuts) are also sometimes necessary.