My setup is identical to the OP's. I do not whirlpool through my plate chiller. I also do not contain my hops in a hop spider (as microbusbrewery suggested) for the exact reason schematix cited: It decreased my hop utilization noticeably. Here's what I do:
My 15 gallon BK has a whirlpool port about 1/3 of the way up (so it's submerged even for 5 gallon batches), and an elbow pickup tube on the outlet port. Hops get thrown directly into the kettle - no sock or spider.
During the boil, I hook up my hoses, pump, and chiller to a small bucket of StarSan and recirculate StarSan through everything to sanitize it. Of course, the pump and hoses will get sanitized during whirlpooling - this is mainly just to sanitize the plate chiller and "COLD WORT OUT" line. I run this for 5 minutes or so, then disconnect and drain everything. I then hook the system up as follows: BK outlet port -> pump -> whirlpool inlet port. At flameout, I add any steeping hops, then I whirlpool for 5 minutes or so, then shut off the pump, put the lid on the kettle, and let it settle for 15-20 minutes. I close the whirlpool port valve, disconnect that line and connect it to "HOT WORT IN" on my plate chiller. Careful, it's hot.
When it's time to chill, I start the flow of chill water, then turn the pump back on, with the cold wort out line feeding a carboy. I have a thermometer and a valve on the "COLD WORT OUT" port of my chiller, so I can throttle the flow of wort to achieve the desired cooling temperature. While this runs, I collect the waste chill water in buckets to use for cleaning later. After I've filled 3 buckets (1 PBW, 2 plain), I just dump the extra chill water down my driveway until the wort level in the kettle reaches the bottom and it starts sucking break and hop trub, then I kill the pump. This takes maybe 5 minutes for a 5 gallon batch.
Cleanup can be annoying, but I use stainless quick-disconnects on my lines, so for an initial blast, I connect my chilling water line to the "COLD WORT OUT" line of my chiller and open my faucet up wide open to blast the inside of the chiller, then do the same thing with my whirlpool port on the kettle. This just gets the bigger pieces. I then give the kettle a quick rinse to get the big particles, then dump in some hot PBW and recirculate it backwards through the chiller and the whirlpool port for 5 minutes or so, then do it again with hot rinse water, then finally one more time with cold rinse water, scrubbing the kettle while this is happening. Then I have everything up to dry.