You won't know until it warms up if any pipes or fittings have broken. Fortunately the pipes are PEX and it is more forgiving of freezing than copper. The water heater tank won't break unless it is full. In fact, it is hard to fully drain and built so a gallon of so won't cause any damage. A light bulb (incandescent) puts off much more heat than light, so one or two under the tanks will provide some warming. If you can drape the outside, it will better contain the heat.
Keep an eye on it as it warms up to see if there are leaks. Once it warms up enough, you can attach city water (keep the pressure low at first) and see if you have leaks—places to watch for include the toilet valve (behind the toilet; shut off valve below it will stop leak for the time being) and the kitchen sprayer, two things novices forget.
If you bought it from a dealer it should have been winterized and you may be able to get them to fix problems. If from a private party, it should have been winterized unless you bought it in southern Florida, and you should contact them about the problem.