Edmonds, WA is staffed for all trains and has checked baggage service. It is the station I use the most.
I hope you know it is LONG way from Kingston to Port Angeles, over 60 miles, and would be a VERY expensive taxi ride.
Since you obviously won't have a car, and so won't need a car ferry, why don't you just take the Victoria Clipper out of Seattle? It isn't particularly cheap, but probably cheaper than that cab ride. I think you can even buy the ticket from Amtrak, I know they sold them at one time, although you can't purchase it as a connection, since the Victoria Clipper dock isn't the same place as King Street Station.
Another, less good option, you can get ticketed directly to Vancouver BC from MSP, which uses a connecting a Thruway bus to get up to Vancouver. There is regular bus service from the train/bus station in Vancouver to Victoria, as frequently as hourly, I think. Either option is probably better than taking a cab from Kingston to Port Angeles.