How do I extend the range of my Airport extreme at home?

You might try purchasing an Airport Express to place downstairs in the Kitchen. Configure it to Extend an existing network. See if that helps improve the range. No guarantee how well it may work because you still have a second floor between the kitchen and the third floor attenuating the signals of both.

It is plugged into an AC outlet downstairs. It is configured using Airport Utility. It does not need to be plugged into anything else downstairs because it is simply connecting to the wireless network created by the AEBS.

You should initially configure the AEX upstairs next to the AEBS. You will need to turn on wireless on your upstairs computer just to configure it. Here are some articles that may be helpful:

Wi-Fi base stations- Setting up and configuring a roaming network (802.11 a/b/g/n)

AirPort Express frequently asked questions (FAQ)

AirPort Express- How to join an existing Wi-Fi network in client mode

AirPort Express- What is client mode?

EDIT:

If the AEX in the kitchen doesn't help enough, then try plugging it in somewhere on the second floor right below where the AEBS is located. You may need to experiment with different locations on the second floor in order to get the best signal down to the kitchen.