In the past, I have used these:
Prestone® Flush N Fill Kit | Print View | Products | Prestone®
If you can get to a heater hose, you can use this to flush the whole system. Be careful, though: City water pressure can be as high as 90 psi, compared to the 16 psi your water pump seals were designed for. So flush the system slowly, just crack the hose valve enough to get the water flowing gently through the overflow of the kit.
When you are finished flushing, you add coolant and push the water out until coolant starts coming out.
Don't touch the bleed screw on a 2.7L engine to bleed the air. It's a metal fitting in a plastic housing and the whole fitting can spin in the plastic when you put a wrench on the screw. Just let the system bleed itself through a couple of warm-up and cool-down cycles and add coolant as necessary.