How much surge protection do external power supplies provide?

There are two kinds of surges, differential-mode and common-mode.

A differential-mode surge means that the voltage difference between line and neutral rises to an abnormal value. This type of surge can be caused by a lightning strike on a distant high-voltage feeder line, which then gets transformed by your local distribution transformer to a correspondingly high voltage at your service entrance.

A common-mode surge means that all of the wires experience the same abnormal voltage at the same time. This type of surge can be caused by a nearby lightning strike, which can cause the "ground" in the vicinity of your house to have a much higher voltage than "ground" much farther away. It can also cause large currents to flow in any wiring "loops" in your house, which also induces common-mode voltage shifts.

So, taking your more general question first, a typical wall-wart power supply will generally protect and/or "take the hit" for any differential surges. There are relatively few mechanisms by which a primary-side overvoltage would be coupled to the DC output. Most power supplies have spark gaps and/or MOVs to make sure that the primary voltage doesn't exceed the isolation rating between primary and secondary. (In fact, I'm fairly certain that such protection — at least up to a certain energy level — is a requirement in order to get a safety rating such as UL.)

However, these mechanisms cannot do anything for common-mode surges. A common-mode surge could easily exceed the isolation rating of the supply and cause the DC connection to the device to also experience a common-mode surge. If the device is otherwise isolated, it might survive this, but TVs (and cable boxes, etc.) tend to have another connection — the signal cable that comes from the antenna or cable company.

Now, the cable shield is supposed to be bonded to the same ground as your AC power at your service entrance, and if this is the case, then this should experience the same common-mode surge as everything else and preventing large currents or voltages from appearing. But if it is not, then large currents can flow through the power supply, the TV and the cable shield. The TV is not likely to survive this.

Also, as I alluded earlier, the path from your power service entrance, through your house wiring to the outlet, through the power supply, the TV and the cable connection forms a large "loop" with a significant amount of area. A lightning strike that's close enough can induce a large common-mode current in this loop even if the cable is properly grounded at the service entrance.

So, in spite of all of that, the bottom line is that no one can say for sure one way or the other whether or not your TV survived. For $15, it's certainly worth a try. If you're an electronics hobbyist (I presume you already have a multimeter of some sort), then investing in an adjustable bench power supply would be worth your while, because in addition to all of its other uses, you could use it to test the TV before committing the money for a new dedicated supply. Units that can produce 0 - 30 V at up to 3 A are readily available at very reasonable prices.