There is a leak coming from the toilet, and will stop leaking after I've shut the water off to the toilet. It will keep leaking until there's no water in the tank and very little water in the bowl. It's to the point where it's a steady drip if the water is left on and you can hear it keep running into the toilet tank. It's dripping into the downstairs kitchen floor (through the purple board that's waterproof) and my landlord won't have anybody come out to fix it. There is no water on the floor of the bathroom when it leaks. What/where is the leak on the toilet and approximately how much/long will it take to fix?

If water leaks out of the bowl you have a cracked bowl. If water is leaking from the tank to the bowl then you also have a leaking flapper or flush valve. However that doesn't matter because the toilet must be replaced. There is no repair for a cracked bowl.
I don't know what waterproof purple board is but rest assured that the ceiling and upper floor joist are being damaged. The lower floor where the leak is dripping is also being damaged. If landlord allows this to continue long enough, he will eventually have to not only replace the toilet, he will have to replace the entire upper floor and the sheathing and the covering of the lower floor.

Landlord must not be very bright. Start looking for new place to live.

I agree with Harold, you are not supposed to repair anything in a rental but the landlord should know that a continual leak will cost him more money than a repair. From what you say this toilet has more than one issue. A crack or poorly made toilet with a porcelien patch would run dry but there would still be water in the tank so your tank flapper must also be bad.