Either the igniter is broke or it's not getting propane. The typical sequence of operation is that the fan should come on, then shortly thereafter (about 3-5 seconds?) the furnace should light. The circuitry requires a heat source for the propane to continue to flow, so if the electric igniter doesn't fire and light the propane to provide the heat source, then the heater cycles off as a safety measure.
Typically, if the furnace stops working in the middle of the night, that's usually the sign that someone (dad) has to crawl out of bed and switch propane tanks. Doesn't appear to be the case this time around though, if the water heater and stove are getting propane.