Are there any tools cheap from eBay that I can use? Or anything common from around the house? And a tutorial.
Do not try to force the peg out using the string, the ball presses against the peg to lock it in place and you can damage your peg slot by forcing it.
Here's how we do it at the shop:
Loosen all the strings, enough to get your hand in the sound hole. If your hand is small enough, reach in the sound hole and locate the end of the pin that is broken. Grab it and gently wiggle it while pushing up.
If it doesn't loosen, I use a small block of wood to gently tap on the peg upwards. You may have to remove the other strings and pegs to get access.
If that doesn't work, you may be able to insert a small stiff wire from the top in the string hole (looking at your picture you may have room). I use a small Allen wrench that is used for electric guitars, or a stiff piece of piano wire and press down on the string ball. You are trying to move the ball end of the string down the peg.
If it is still stuck, then you will need a small drill bit, smaller that the peg hole opening. Carefully drill out the plastic a little ways down and then try working the peg up from the inside again.
Similar question: What is the best way to remove a broken bridge pin?
Have you tried loosening the string yet? Most pegs are designed to slip up and out when the string tension is released. Once loosened, try pushing the string down into the hole by gripping it as near to the bridge as you can and giving it a wiggle. If it still doesn't budge, follow the previous post's instructions by taking all of the strings off and attempting to push it upwards from inside. I've successfully used the back of a long wooden spoon in the past.
An alternative to drilling - should you need to go that far - might be to find a small, sharp, self-tapping screw that you could work into the remaining plastic and then pull upwards, being extra careful not to screw into the wood of the bridge.
You'll need a new peg anyway, regardless of how you get the old one out.