This is weird... I went back out to the car just there and the new key wouldn't start the engine anymore, so I thought that I'd try it one last time at re-programming it. When I press the unlock button the second time, the car alarm sounds. It is then able to lock and unlock the car, but still unable to start the engine...
I think some people are getting a little confused here. You
can't just buy a new key and expect it to start your car after getting the blade cut. If you are not using the transponder from an old key
that is already matched to your immobiliser, you will need your Security Key Code (SKC) in order to program your immobiliser to accept the new key using VAG-COM. If you don't have the SKC for your car, you will either need to find somebody with VAG-Tacho to get it from the car for you, or you will need to pay a SEAT dealer to synchronise the new key for you.
You can only manually synchronise the remote locking part of the key to suit your car. You cannot get your immobiliser to accept a new key with a new transponder without knowing your SKC and getting VAG-COM to do it. The reason you were able to start your engine before is that you had used the old key in the ignition immediately before using the new one, thats why the immobiliser light was flashing. This works but it only works immediately after using a matched key.
When I got the car I only had one key, so I bought a second flip key from eBay, then I bought a replacement blade which I got cut the other day.
Will the second key need a new transponder? I was under the impression that the one that is already in it would be able to be re-programmed to my car?
Its not a matter of re-programming the transponder in the key, you can't do that. A transponder of the type used in Leons has no capacity to be programmed, it transmits a set code when interrogated by your immobiliser. What you actually do is program your immobiliser to accept that code.
Cars built after 2002 were not supplied with an SKC. If this is the case, your only option is VAG-Tacho or a SEAT dealer.