Hi there.
Firstly if you are looking to book a hotel in Funchal then be very careful. Most of the hotels are situated in what they call 'the hotel zone' which is a long way out of Funchal centre.
I am not sure what your budget is but friends of ours have stayed at The Cliff Bay 3 times and it certainly seems to tick all your boxes. However I think it is 5 star and therefore quite expensive! The hotel is part of a group and I think they have other hotels in the centre which may be cheaper.
We have been to Madeira independantly and so have our friends. We have both used BA from Gatwick.
For a package look at Strawberry World -
http://www.strawberry-world.net
There is a lot to look at strolling along the fron in Funchal - not least the cruise liners that come in. Remember though that there are no beaches.
If you require any more info then please ask.
Pippa