For the filter - your main pain point will be all the bolts and nuts holding the lid on. Obviously, have everything off before you start the deep clean. Open the air valve, with all the skimmer/jet valves in normal operating condition, and let the filter self drain back to the pool. Then work on the lid. Take care on opening - it may be gummed in place and take some convincing - no tools that could damage the seal (likely a giant o-ring). Get out the hose, and work it up and down in the entire sand bed. You will be flabbergasted at the amount of dirty water that comes out. If the sand level is low, add it before starting the clean - new sand has a lot of dust, but the deep clean will rinse that out too. If there are very hard chunks or matted together things, I have been know to just get them out and say good bye. Keep working it until you are convinced you're good and the water overflows clear. Make sure there is no dirt/sand in the seat for the seal, and the seal is in good condition. Pool lube on the seal, if rubber, is a good idea (another messy job). When putting the bolts back in, do it like a tire, with one on one side, then on the opposite side, working your way around.
Afterwards, test your setup for the pump speed issue. It may have been caked together sand causing the problem.
IF pool clearing still seems slower, you can add DE to help it along.
Add DE to a Sand Filter
Sounds counter intuitive - but a slightly dirty sand filter filters better. The difference is that the DE (or new accumulated dirt) will backwash out pretty easily, and not plug things up. The only downside to DE is that sometimes you have to purchase what for you would be a multi lifetime supply.
And pour yourself that glass of patience after. Run your pump at a little lower speed, rather than full bore, so that it doesn't just blow the dirt right through the sand and back into the pool. That was my issue in my sand days - too much pump on a too small sand filter.