In addition to a well stocked kit, you'll get a lot of mileage out of better training. The basic First Aid from the Red Cross is a start, but, for longer term scenarios, you'd be better served with a Wilderness First Aid, or Wilderness First Responder course. (Ok, there are Wilderness EMT courses too if you've got the time/energy, it's a lot of work.)

I know, you're not in the wilderness. The thing that makes these courses good for the scenario you described is that they are designed for situations where you might not have the EMS system able to reach you quickly. They are designed to teach you how to handle medical problems when more advanced medical help may be days away. Most of the folks I see at these are outdoor group leaders, outfitters, hunters, Scoutmasters, etc. Still, the knowledge would be valuable even in non-wilderness situations for handling disaster scenarios where the Fire/Police/Rescue services are overwhelmed. Some examples of places that do a good job are:
National Outdoor Leadership School's Wilderness Medicine Institute, (NOLS WMI):
http://www.nols.edu/wmi/

Stonehearth Open Learning Opportunities (SOLO):
http://www.soloschools.com/index.html

Good luck!