Gaggia Classic - water not flowing

Hello there.
So water isn't flowing through my Gaggia Classic. I know I've been bad and not taken good enough care of it to let it get into this state.
I called Gaggia who reckon it just needs descaling, and will charge me 80 quid to repair it including postage.
The usual descaling procedure won't work, as I can't flush a descaling solution through the system.
I've removed the top of the machine and can see that when I turn the pump on, water comes up one of the tubes just about 2cms, then back down, and back and forth. It's trying to **** water up but it can't.
The other tube (that starts in the middle of the machine) looks like it has water inside it, near where it connects to the rest of the piping. 
Any suggestions? I was thinking:

- pull the the two pipes up out of the holes that lead to the water tank and hold them up vertically - turn the whole machine upside down, so that all the water inside the pipes drains out of the pipes - make a 4x normal strength descaling solution - funnel a small amount into each pipe - wait 5 minutes - repeat all the above - then try to do the normal descaling-run-a-solution-through-the-machine
Any help appreciated!

Cheers, -Phil