Seattle doesn't have an NHL team because when the old Seattle Coliseum was gutted back in the mid-90s and rebuilt into Key Arena, the Seattle Sonics (NBA) and the city of Seattle decided that the arena's seating would be configured around a basketball court, instead of around an NHL-sized ice rink.
When the Seattle Thunderbirds (WHL) play at Key Arena, they have to close off one end of the arena's seating just to get the rink in. They can seat 15,177 for hockey, but with obscured sight lines in one end.
When the Hartford Whalers were looking to move in the mid-90s, Seattle couldn't take them in because Key Arena's seating didn't fit the ice rink and didn't have enough luxury boxes for an NHL arena. So essentially, there is no arena in Seattle that's built for NHL hockey. There has been talk about building one, but it's just been that - talk.
After the renovation, I personally couldn't stand to go watch games in Key Arena because of the botched seating arrangement. It just pissed me off too much. I have many fond memories of the old Seattle Coliseum, and Mercer Arena, too.
The closest place you could find to Seattle, and still be in the US, would be the Portland Rose Garden Arena where the Portland Winterhawks (WHL) sometimes play. They can seat 17,500 for hockey. Owner Paul Allen would like to bring in NHL team there, but hasn't been able to as of yet.
Portland's probably the best place I've ever watched a hockey game - I was at the old Memorial Coliseum, tho'. The fans are great, and I think they really deserve a team there.