IMHO pets are quite simply the best therapy.

Whether your landlord would be sympathetic to your request will depend on lots of things, how long is your lease, how many other tenants does he have, has he other tenants with service animals, has he animals of his own? Try to think what his concerns will be and come up with some suggestions as to how you might deal with them. If you can present a case that minimises the impact of some of his concerns then you will stand a better chance of gaining his willing co-operation. Rather than asking the question outright, I'd suggest testing the water first, if he is openly hostile then it is up to you how far you want to take it, however if he is sympathetic then you'll know its worth asking outright. Good luck.