You will not be able to see Illuminations from the Doubletree. You may from Buena Vista Palace... but not completely sure. We Stayed at the BVP, had a "fireworks view" which we could see the fireworks from Magic Kingdom waaaay off in the distance. From a different side of the hotel, perhaps you can see Illuminations.
Staying at the Dolphin, we had what was called a deluxe corner room at the time. This room was in one of the four wings that stick out towards the Swan.
These are great rooms, some of which have a balcony. From the higher floors you can see some of the fireworks and lasers from Epcot, however a majority of the show is low to the ground. We were on the seventh floor and had a pretty good view. It was enjoyable, but only a fraction of the show could be seen.