Communicate your requirements to your broker. With the occupancy rates what they are, a landlord might be likely to include this in the cost of the buildout for occupancy. Most buildings are not going to be equipped to handle that sort of cooling with their current system. I think you are severely underestimating your heat load.
A person is generally considered to be 400btu. Even some of the smaller Dell 1U servers claim 850-1400 BTU as a minimum. It wouldn't take too many servers to overrun 5000 BTU of cooling. I can tell you that the data center where we are complains of our heat bloom and has put in much higher airflow tiles in the floor. Our current generation servers are dual quadcore Xeons, green certified, 90% efficiency power supply, and they are rated at 1300 BTU. A rack of those not including the support hardware (switches, etc) is 41.6k BTU.
1 watt = 3.4(1something) BTU if I remember the math correctly.
In NYC, you might be able to get hooked into the buildings chilled water supply to put in a fairly small but efficient AC unit. Remember that an air conditioner moves heat from one place to the other. Where you are providing cool air into your closet, that hot air needs to be exhausted somewhere either in the plenum, through ducting outside the building, into the neighboring office space (if you don't like them and they are a competing webapp company).
Recalculate your heat load, have your broker make sure that the space will have a server closet with adequate cooling, or, ask for a concession and build it yourself with a qualified contractor. Replacing hardware that overheated is a frustrating process as the units will not fail in predictable manners.