I've had success on other cars doing the following:
Get a small submersible pump like a Little Giant or equivalent. Get a 5 gallon bucket. Make some hoses long enough to go from the firewall heater core inlet and outlet to wherever you can put the bucket. Hook one to the inlet and one to the outlet and drop into the bucket.
Now get the hottest water you can and put in the bucket, along with 2 - 3 bottles of PRESTONE RADIATOR FLUSH. Hook up one hose (doesn't really matter if inlet or outlet) to the pump and drop into your solution, plug in and let it circulate for 24 hours.
When I've done this in the past, often times one will get next to no flow coming thru the system, you'll have a lot of backpressure on the pump side, so make sure you have some way to allow the backpressure to bypass the pump, by either using a pump that allows bypass or simply having the hose on the pump put on loosely to where some pressure can escape. You can also alternate between having the inlet or outlet hooked to your pump to maximize the cleaning ability of the flush.
If you achieve the same results I did, you'll be amazed at the amount of 'gunk' that comes out. Once you are satisfied with the results and have decent flow thru the system, run a couple of buckets of clear water thru to get out all the radiator flush chemical.
2010 Ingot Silver FX4 SuperCrew