How to dial a phone number via DTMF tones?
I tried soldering a 3.5mm audio jack to a phone cable and using my regular phone app, but nothing happened.

You can consider yourself lucky. The PSTN system works on 50 V DC and the ringing voltage can be up to 100 V AC (20 Hz, I think). Both are enough to destroy your phone if you connect your phone to a telephone socket (which is what you seem to be describing).

In this video, this is done by simply playing the tones in to the speaker, but that wouldn't work on my phones either.

You haven't described your landline system. If you hear DTMF when you press the keypad then it is most likely to use them for dialing and playing DTMF on your mobile phone's speaker while held up to the landline microphone (not the speaker which is the earpiece) should work.