By LP or HP, he probably means low profile or high profile, meaning how high out of the slot does it sit. If you have high sitting RAM, typically due to chunky heat spreaders, they can block some heatsinks.
Personally, I use a
Scythe Fuma Rev B with a triple fan setup on my new build, but you don't necessarily need a new cooler. I'd start with simply reapplying thermal compound. I prefer Thermal Grizzly Kryonaut, but that can be a bit pricey. Any quality TIM will work, whether Arctic Silver 5, Arctic MX-4, Noctua NT-H1, etc, etc. Frankly, the difference between top end and bottom end TIM is a handful of degrees at most.
Also, the only mention I've seen of temperatures was when you mentioned 80°C during a stress test. It's been a while since I have worked with Sandy Bridge, but I don't think that's horribly hot for a heat-generating stress test (IBT, especially with AVX; Prime Small FFTs, AIDA64 with FPU, etc). I could be wrong, though.
[Edit: Found a thread showing that a 2600k will thermal throttle at 98°C, so a 2700k should be the same. 80°C shouldn't be terrible for a fairly short term stress test.]