DVD Cd drive doesn't read disks

There are a few ways to try & get an optical drive to read disc media;

one is to buy or otherwise obtain a cleaning disc. Some aren't good.

Avoid discs with brooms that hit internal components.

A few may work OK. Some users have tried a credit card or thin piece

of plastic and put a bit of cloth over it, such as a microfiber lens cleaner

& carefully insert then remove the cloth (over stiffener plastic) to clean.

suggestions appear from a search online such as this; also see comments:

http://hints.macworld.com/article.php?story=20120210114733216

Sometimes, a few suggestions have included air, carefully blown into the

optical drive opening, past the dust cover inside, to see if any loose dust

may move from the path where the optical laser lives. This can be the

cause of a problem if you were to say, use a canned-air product that has

actual liquid content that turns into 'air' on release; when those cans are

tipped, a cold liquid may spray out and that can damage components. It

also may freeze fingers. Not all canned air product are equal or desired.

• Get help with the slot-loading SuperDrive on your Mac computer - Apple Support

• Mac Mini repair guides (all models) - iFixit:

https://www.ifixit.com/Device/Mac_Mini

The unit, the optical drive, may have failed. You could probably replace

the combo or superdrive yourself if you can follow instructions from one

of the online guides to repair the model Mini you have, without causing

additional issues; since accidental damages can't help the situation any.

Does the optical drive appear in the Disk Utility? It should. Also there

may be a listing of what the system sees, in hardware & software, in

the 'system profile' or hardware listed from About This Mac, more info.

Replacement drives may be located by looking into the online sites of

Mac-centric vendors of good reputation, such as OWC macsales.com.

Compare specification and part numbers of quality brands, and see if

you can perform the operation yourself; or contact authorized repairer

in your area and ask what their fee would be. A diagnostic may be an

advisable choice before an actual repair; unless you can determine the

unit (optical drive) is indeed defective without a doubt.

Good luck & happy computing! 🙂

Have you tried a cleaning disk?

Also, can you boot form an install disk?

If not then the drive failed. I would use and external optical drive rather than replacing the internal optical drive. An external drive powered by an AC wall adopter is best since the USB ports on your old Mac are note USB 3 and thus have problems putting out enough power for an drive powered by the USB bus.

You can also then replace the internal optical drive with another storage drive (HD or SSD)