This must be your first cruise otherwise you would know the answer to your question. My advice is to try a short 4 or 5 day cruise with RCCL from Barcelona during May/June on Navigator. Taking a month cruise can become boring and if some things on the ship are not to your liking, the thoughts of having to tolerate them for so long can turn a dream holiday into a nightmare - especially involving one so young.
Holland America " Zuiderdam" do a fantastic 24 night cruise (europe) on the 31st July. Barcelona to Venice.
Good value cruise for august.
P&O have great facilities for kids and you get a futher discount if you book 2 two-week cruises back-to-back. The itineraries vary so you can sometimes pick two cruises with different ports of call. Unfortunately, as most of the cruises go from Southampton you end up returning to Southampton mid cruise. Princess also have good facitilities for children and have ships based in the Med so again you can book cruises back to back to make the full month you are looking for. Although I haven't sailed with them, I have heard that Royal Caribbean are also noted for their kids facilities (you can even leave your child on board whilst you go ashore if you wish) and again they have ships based in the Med throughout the summer. If you don't mind flying longhaul there are lots of back-to-back itineraries for the Caribbean although I have heard that some people find the week long circuit there quite enough so you might get a bit stir crazy if you spent a whole month in the region. Hope this helps.
P & O are very friendly on the ships which caerry children. Note not the whole fleet carrry childre.
We find P & O have very good Kids clubs , Our daughter was 3 and didn,t want to leave the club on Oriana, Everytime we asked where did she want to go she said Peter pans. The clubs are open from 9.30 - 9.30. They have a night nursery too. We could leave her at the club and enjoy a peaceful evening meal. They put on Kids tea at 5.15pm They youth crew were very good.
Royal Caribbean do kids brilliantly but not sure if your daughter could go into the kids club at 2. P and o are also getting more kid friendly. As far as eating goes you might opt for a ship with a 'buffet style' option or freestyle dining as well as the formal option (especially for a month). Most ships will do anything you ask for if your little one is fussy.
Bear in mind that most ships do not allow nappies of any description in the pool.
Cabin sizes , ie. square footage, are freely available on the internet so you can consider in advance what life in the cabin will be like for a month. We've done sofa beds, pullman (fold down from ceiling) bed and connecting cabins all with success but with a 2 year old you will need a sofa bed - once it's down at night some cabins are a bit tight to get around in. Most ships have good storage in the rooms but forums on this site and cruisecritic.com will give you lots more info.
As far as destination is concerned the world is your oyster! The caribbean could be windy and wet but might not be. The med is a good choice in July August but the ships have many more children on them. Some big ships sail from Southampton to the Med now (eg Independence of the seas)
Budgetwise if you can travel out of season you can get better deals and also the Caribbean is more economical sometimes even when you take into consideration the airfare!
(innoculations / malaria requirements are another consideration with a little one).
Hope that's usefu.