Using old IMac as screen extender with Macbook

What you are trying to do, the now-deprecated Target Display Mode (TDM), won't work with that combo of Macs. The very restrictive hardware and OS requirements for TDM are outlined here:

Use your iMac as a display with target display mode - Apple Support

Your 2011 iMac is within the TDM reqs, but a "new" MacBook (2020 or newer) is not capable of using an iMac as as second monitor via TDM. There are aftermarket software and "dongle" options but they tend to be pricey and all cause lagginess.

The IMac has the ability to mirror to my MacBook using airplay, but with this I can only use the IMac as my main control, not control the IMac with the MacBook

Your iMac is too old for Airplay to Mac to work like you want. See:

Continuity features and requirements for Apple devices - Apple Support

Apple began the deprecation of Target Display Mode in late 2014 and tossed the last shovel of dirt in its face in early 2020, so it dead tech.

For cost, ease of set-up, fast performance, and full display options, the only viable option today is a free-standing external monitor. That will also use less power. You can buy a very acceptable 20-24 inch monitor for only a bit more the some of those aftermarket options I mentioned.