I've used window units for cooling one room, and they work well for that. I have central air, but the upstairs bedroom needs a window unit to help get the temperature down at bedtime.
Air conditioning is more than just temperature control. Humidity control is the other half of the equation. Once the whole house is dried out inside, higher temperatures will feel more comfortable. Here's a chart that shows this:
http://epb.apogee.net/res/refcomf.asp
Consider getting an HVAC technician to diagnose what's wrong with the central air unit. It may have just lost its charge for some reason, and that's not necessarily difficult or costly to repair. Once you get it running, life will be more pleasant.