Hot water stinks, but cold water is fine

We have a small hot water heater in our office. The hot water smells like rotten eggs at the sink when you wash your hands. We have drained it and run lots of water but the smell remains. Should we flush it with white vinegar?

Pull the anode rod. But if its material has broken off and is laying in the bottom of the tank, it may not be correctable without replacing the water heater.

The metal in some anode rods can react with some water to produce that smell. The choice is to either remove the anode rod as suggested, or to replace it with one that won't react to the crud in your water. To get it out, you may need an impact wrench, since they are in there tight, and it's been there awhile. the anode rod is more technically called a sacrificial anode, and it is the 'weakest' thing in the tank, and gets eaten up first, rather than the tank rusting out. So, removing it can speed up the demise of the tank.