The Sofitel at Yerminal 5 is lovely, good restaurants, a tea room too

There is a Sofitel at T-5 (very pricey) and a Hilton Garden Inn (less pricey) at T-2 which is walking distance from T-3. A T-4 ( a slog but free transport to/from T5 and T3) there is a yotel and Hilton. The Yotel charges by the hour so might be a good option.

Are you flying Business or have lounge access. Most of the Lounges have shower facilities.

Yes -- that would be the absolute easiest -- not sure what they charge for a day room. I wanted to stay there my last trip before flying home out of T-3 but only the suites/pricier level rooms were available so I stayed out on the Bath Road like usual.- for 1/3 the cost. For my next visit In early Dec the Hilton Garden Inn's rates are high for my specific date so I'm staying at the Hilton at T-4 for nearly £100 less. Less convenient but my flight isn't til 11:30AM so a 20 minute travel time is no big deal. Hope to stay at the Garden Inn sometime soon though.

We stayed at the Sofitel only because we were flying out of T5 and had just enough time for a jaunt into London. We highly recommend.
Otherwise, our first choice would have been the Hilton Garden Inn at T2.