Silvia, as dreadful as it is, it is not considered a valid reason to break your lease or not pay your rent. Unfortunately, cities like New York, Boston, & Chicago are plagued with them. Legally, your landlord doesn't have an obligation to remove them. There are too many pigeons and there will always be more to take their place. Now, if someone is keeping pigeons on the roof illegally or feeding them, she has a responsibility to put a stop to that.
There are some things you can do. If it is just that one room that is a major problem, there are some methods you can use to scare away them away. Some people use plastic owls that are sold in hardware stores but city pigeons are tough and I am not sure if once they see the owl not moving, it may not have much of an effect. There are other methods as well. Here are some links on pigeon pest control. One of them sells an owl that moves which might just do the trick. Please review the links carefully to figure out what method you would prefer to try.
Birds: Pigeons, Rock Doves, Pigeon Elimination
Solving Problems with Pigeons | The Humane Society of the United States
Pest pigeon repeller to scare pigeon pests away - Prowler Owl life-sized owl decoy pest pigeon repeller.
You can also point out to your landlord that pigeon droppings are very damaging to buildings. If she is at all interested in maintaining her property, she will call in a specialist. The last resort would be to tell her that it is just too much for you to take and ask her nicely to let you out of the lease. See what she says. Always try the pleasant and direct approach in an unpleasant situation. She may be very willing to let you out of the lease.
P.S. Hullo Tickle! Long time no see. Didn't see your post when I posted.
P.P.S. Silvia, you can always call the Health Department to inspect the premises. If it is as bad as you say, they will tell her she has a certain period to remediate the problem otherwise she will be fined. But, be forewarned, what you consider a health hazard, they may not.