Hello @zdravke ,
You wrote: “Only thing Ring needs to do, is put out zero resistance at the terminals to ring the chime, but instead it remains at 240K.”
Well, that would be true IF your doorbell utilized a mechanical relay, but it doesn’t.
You can not simply treat the Solid-State Relay in your doorbell, like a mechanical relay, and use a meter to check if it completes the circuit.
_“Solid-state devices require a minimum amount of voltage/current across the input before the output device will conduct. Most multi-meters are designed to simply provide a small amount of voltage in order to measure the ensuing current draw. Therefore, when measuring a solid-state device, the meter will typically report an open circuit (or very high impedance) because it cannot put out enough to turn on the relay” _ https://forum.digikey.com/t/can-i-test-solid-state-relays-ssrs-with-a-multi-meter/546
This is why you are getting the ohm meter reading of 240K (the very high impedance).
Since your current house-bell does “Ding-Dong” chime when you “short the wires”, but then does not work with your Video Doorbell 2 (with the “Mechanical” setting mode), this means that @Caitlyn_Ring is correct, and that your old house-bell does not work because it IS NOT compatible with your Video Doorbell 2. If you simply replace your old bell with an inexpensive, newer bell-chime that is on the compatible list: https://support.ring.com/hc/en-us/articles/115003008663-Ring-Video-Doorbell-2-Chime-Kit-Compatibility-List then your Doorbell 2 will be able to “Ding-Dong” a new compatible house-chime-bell, just like your old bell did, in case your Wi-Fi goes down. 
Additionally, I too do own a Ring Video Doorbell 2 and it does make my house wall-mounted bell go “Ding-Dong” just fine. Here is more detailed information explaining the differences between mechanic relays and solid-state replays, and why using your meter in a typical manner doesn’t work:
http://www.crydom.com/en/tech/newsletters/solid%20statements%20-%20bench%20test%20an%20ssr.pdf
I hope you find this information helpful 