Is there a shut of valve under the tap? If not you will need to shut the whole house off to fix it. What kind of tap style is it? Mono mixer (one spout one tap to move like a lever that mixes hot and cold) or two separate taps that turn one hot one cold that just mix into the sink or a common spout?
It is one tap and it needs to be replaced. That is the problem with renting facilities - you get second rate stuff. I like the tap that has a lever... I have found that the turn types ware away too quickly and often give trouble because they are either turned off too tightly or not enough.
You know your taps JB. Too tight wreak it and drips cause wearing. I broke a tap last night will have to dissect it, the spout is suppose to swivel but to suddenly grabbed and the whole thing turned! Now we get some water leaking out the side.... I assume that something has jumped out of place...
Maybe a little more expensive, but I also find brass being the best way to go for taps rather than metal or plastic. They handle constant abuse better and because corrosion is almost nil you will find that parts continue to hold better - therefore no leaks. The key thing is to ensure that the WHOLE tap is brass as some try to sell you brass casing with inner parts that are plastic or metal.
The tap that has a sensor for auto shutoff is great too - saves water and I have never seen one of those leak even though they are used so often in public places where usage is high.