Here is a 5 KHz test tone on Youtube--that should easily be played by just the tweeter. Not exactly musical and it sure don't have a beat you can dance to but will tell you what is what. Basically, play that tone at low volume and put your head holes right near the tweeter. If it is not making any sound, it is blown. If it does make sound, it might be partially damaged or work fine. If it is loud and clear, sounds like your other speakers then you are OK.
The BEST way to know is to remove the driver and ohm it out. It should give a DCR of 6.5 ohms or close. Pull the diaphram off and sniff it--sniff it like a cocaine cowboy and if it smells burnt--it is damaged. If it don't smell burned, the coil is not charred black and it ohms out fine--you have another problem. Have no idea about Klipsch crossovers, they might have a self-resetting breaker in the crossover board--common for pro sound speakers but not common in consumer speakers. You also can have a damaged crossover, this can be caused by dropping the speaker which can pull loose heavy iductor coils. Not common but it can happen--look at the crossover to see if anything is loose on the board. You also can have bad solder points and so on--how to check that! Basically, get a full range driver around 3 to 5 inches and connect it to the crossover. Make sure the speaker is around 8 ohms itself. It should produce highs of some form when playing the 5 KHz tone. If it does not, then you have a damaged crossover, bad connections or even bad capacitors or resistors (it happens)
Check if you hear 5 KHz with the test tone--get your ear close, or if you can play it really low in volume--place your ear on the horn. (be careful!)
If it don't play, remove the driver and smell it for the tell tail burnt electronics smell... if it stinks--that is not a good sign.
Ohm out the driver, it should be 5 to 8 ohms--if it is something like 2 ohms the voice coil wtarted to melt and the coils started shorting together. If it is over 10 ohms, it is toast--replace part.
If the thing smells clean, the coil looks copper colored instead of charred black and ohms out properly with a meter--then you have a crossover/wiring problem. Do the 8 ohm speaker test with the 5 KHz tone and listen for any output.
Fairly simple ans straight forward, very easy if you have a multi-meter laying around. Be aware the "battery test" should not be used on tweeters--you can blow them doing that. Cracking a 1.5 volt or 9 volt battery across midrange or woofers is fine, tweeters can be damaged from DC so best to use a meter.
I learned plenty about blown speakers--my friends are idiots and I love the smell of burnt voice coils in the morning--smells like lessons learned.
Good luck in your quest and I hope it works out well for you.