Firstly and you have to do this yourself before anything else. Confirm that it is the water coming off the paving, be 110% sure.
Just go around the house with a long, accurate level and check what the water on the paving is doing. If you don't have one, I suggest you get a long Stabila, failing that you could spray a lot of water on the paving and see where it runs. You'll need a level for laying the pipe to the soakaway anyway and Stabila are worth buying, as they are accurate.
Pump: Gravity is your friend, it's free, cheap to run, easy to install, quiet, never breaks down nor goes wrong. No pump can beat that unless you want water to run uphill.
Soakaway: Building regs say 5 meters away from the house (or is it 4m, can't remember precisely, but something like that). The drain needs to fall the right way and if you have lower ground, it's less digging to put it there. You can do all this yourself and save thousands to spend on a nice holiday instead.
A deeper hole will work, as long as the pipe isn't laid flat and connect into the bottom of the soakaway, I'm sure somebody must have tried that before.
You're right Canardly. I think it was common in the 80's/90's to still have suspended timber floors. Whoever laid the path should have dug it out to be lower than the ground inside the extension or built the extension higher than it is, but then you'll have a step up inside the house.