Can we power the monitor part with a new Mac mini?

Your answer is in the link that was provided. In short, it is simply too new. Also, the 2010 27" iMac must be in perfect working order for Target Display Mode to work on pre-2019 MM's and the MM has to be running versions of Mac OS Catalina or earlier. In other words a "dying" 2010 simply won't work. If you would like to recycle the 2010, you may want to take advantage of Apple's free recycling program. You can find out about it at Apple Recycling Programs

To be honest, it is ALWAYS best to buy an external display, why:

• Displays are available <$100
• No expensive cabling necessary
• Simple to setup
• Dead reliable

If the new Mac mini is a model from 2019 or earlier, maybe. Otherwise, no.

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

Note that 21.5" 2010 iMacs never supported Target Display Mode at all. Only the 27" 2010 ones did.

The other option would be to use third party software like Luna Display or Duet Display. These options work best if both computers are connected via an Ethernet cable.

Hope this helps.

Paul