Two weeks doesn't seem like an inordinately long time for a true double IPA. Also depending on the OG 1.019 might be the bottom. As this was an extract beer I'd be surprised to see a double IPA go to 1.010; unless this is something you've brewed before and gotten to that point.
I'd say warm it up to 70-72F and give it another week.
Stirring, IMO, is a bad idea as it will introduce some oxygen. While that might be great for the yeast it might be bad for the beer.
If you don't mind sacrificing some beer for you own education. Pull off on gallon and split it into two sanitized containers. To one go ahead and swirl it, get some oxygen into it. Let it ferment another week and see if it goes lower than 1.019. That will show you the effect of more O2 in the beer.
In the second half gallon, add a half packet of dried yeast and swirl the thing every couple days for a week keeping it as warm as possible. This is known as a forced fermentation test.
Next week check the gravity of the two half gallon samples and the main batch. If the sample with the extra yeast has fermented down then you know there is still fermentables in there that CAN be fermented out. Hopefully the main batch will have moved along some too.