The main problem is you need a way of adjusting the chain tension.
Either you need slotted dropouts or a chain tensioning device.

Probably not worth the effort of fixing the back wheel you already have.

Fixed hubs are readily available. You will almost certainly need one which is 135mm over-lock-nuts (OLN). Track bikes are 120mm and ordinary road bikes are 130mm so you don’t want those sizes.

It’ll only work if you have sloping dropouts for your rear wheel, so that you can adjust the chain tension by moving the wheel itself.

You might need to change your chainset and bottom bracket to get the chainline right, though.

There is a lot of good stuff about building a fixie on (the late) Sheldon Brown’s website.

The cheapest way is to remove the rear freewheel and both derailleurs, fit a single cog, respace the axle and have the rear wheel redished by a cycle shop. You will also need to either replace or shorten your chain. You can use the original chainset, although obviously you will only be able to use one of the rings – probably the middle one on a triple.