From Bath Road to T2/T3 is about a £10 taxi fare, plus £5 drop-off charge which is imposed by LHR. You won't have to pay the drop-off charge personally, the taxi driver will do that and add it to the fare. There's no issue with that length of journey.

If - as sounds likely - you don't have vast amounts of luggage with you, the standard TfL (Transport for London) buses will cost you £1.75 each, and you get one going down to Bath Road to Central Bus Station and vv every few minutes, including through the night (albeit less frequently, but still at an acceptable level for most people). Numbers 105, 111, 285, plus the N140 and N9 at night work. If you tell us what hotel you've picked we can work out which stop you need in both directions (they may vary due to having to cross the central reservation).

It would also be helpful to know which terminal you're leaving from at LHR, because that can change a lot of the advice in terms of how best to get around.