We have a very old peace lilly in our office that is quite tempermental. It seems as though we can't get the watering just right for it and we have recently noticed hundreds of little bugs in the soil. Do you know what these might be and how do we get rid of them?

The pest you have might be what's called, "Springtails" They can be seen scurrying around the top soil and in and out of the soil and hopping or "springing" up a little when the soil is disturbed or watered. They will not harm the plant though, they feed on the decomposing material in moist soil.
They might have been there when you bought it and have since become invasive over time.
To keep their numbers down, next time you water, add a little mild dish soap to the water so it's slightly soapy, or, hold the plant firmly in the pot and turn the pot upside down outside or over a garbage can and let the soil fall out around the roots, you may have to loosen it first without disturbing the roots. Then add new soil.
When you water, water the entire rootball thoroughly when the top inch or two has dried first. Empty any excess water in the bottom saucer so its not sitting in it which can cause root rot.
You want the soil to always be a little on the moist side, but not wet all of the time. A once a week watering is all I do for mine.