Split your army into two: the units you start with in the mission + the ones that eventually join you, and a group of stalkers + colossus. As soon as the second army has a decent amount of units in it (shouldn't take long), grab the mine on the left and use it up asap, with the stalker+colossus army defending it.
The other units should just bounce back and forth between the other two entrances. Keep them roughly evenly sized till you hit 200. When you start running out of mineral nodes, you can send your probes to their deaths to get more combat units worth of supply.
At that point it's just a matter of defending your entrances losing as few units as possible per attack. You should be able to defend 2 entrances at the same time. Always have ~3 phoenix around in each army to gravity the hybrids, rebuild any time supply goes under 200 and max out upgrades as soon as possible. You just have to keep at it for a while. At one point I had nothing to do with the minerals I had stockpiled, so I just littered all of the inner walls of my base with a line cannons, and completely filled the two raised platforms, and it really helped thin out the attacks when they came in a line instead of massing their units together.