Trying to fix broken capacitor, help?
I'm afraid it's not so easy.

The fact that you've found a blown capacitor doesn't mean it's the only faulty component, nor even that it's the root cause of a failure that may have destroyed several components. It could have happened that another component may have been the root cause, and then its failure may have propagated down to the capacitor and blown it up immediately afterwards (for example, by delivering a voltage above maximum rating to it).

Each kind of component has its own failure modes. Some devices fail in short-circuit mode and get burned, making their failures pretty obvious. But others may fail in rather silent ways that can't be spotted with a simple visual inspection.

So, if you're not a qualified TV repair technician then your only hope is trying to get a service manual of that TV model (google is your friend) and navigate through it until you detect all failed components and isolate the root cause. Then you will find what's the actual difficulty of the repair, and be aware it can range from something as easy as just having to rework some solder joints, to having to replace an expensive and hard-to-rework IC.

Good luck!

PS: Your replacement capacitor looks like it has seen better times. Did you test it before soldering it in place?