You didn't say how much power your Peltier element needs. And if you're only using a single Peltier device, and running it at full power doesn't raise the temperature enough, then you probably need two, or possibly three devices. That's pretty important to know before continuing with anything specific to use as a controller.
Here's a link to a project that could be a starting place for you to consider. It uses standard, reasonably cost, readily available components. You can get all the components in through-hole packages (not SMD) from Digikey, so no difficult soldering needs be done.
According the the write-up, it's capable of handling up to 100W of devices, so if you need more than two 50W Peltiers, then a more powerful controller would be necessary.
Cheers,
DaveM