Very very close to the flower shower is Belmond Cadogan hotel (Sloane Street) and No 11 Cadogan Gardens and the Draycott Hotel (also on Cadogan Gardens). They are pricey but really beautiful and so close and the atmosphere will be all abuzz with Chelsea flower show.
There are less pricey options - Zed Hotel very close to Victoria Station - a 10 minute nice walk through elegant Belgravia to the flower show.
There are a few hotels on Ebury Street in belgravia which might be worth looking up.
There is Jumirah Carlton Towers, on Sloane Street which is currently being refurbished but due to open ahead of the flower show.
The Millennium Hotel on Sloane Street is not especially special but is a very good location and will have very good views.
Some web sites of areas near the flower show
http://Www.dukeofyorksquare.com and http://www.sloanestreet.com
For the week of the show, many of the local shops will dress up their shop fronts (www.chelseainbloom.com)
and there will be a buzz around the whole area with some stores doing promotions.
There are also hotels at Chelsea Harbour but that is slightly off the beaten track.
I know the area well so I would be very pleased to answer any questions.
Happy planning!