Why won't these two BJTs connected in series work?
I think the main problem is that your circuit isn't DC-biased near the middle of its operating range.  Your darlington drive requires two junction drops of DC on the input before anything happens.  However, a small voltage change on the base of Q1 can cause a large voltage change accross the speaker.  As a result, you can't really bias this circuit reliably without some feedback.

The simplest way to add bias is to put a voltage divider from the top of the the speaker to the base of Q1.  Figure the base of Q1 needs to be around 1.3 V.  You don't show what the power voltage is, so I'm going to do the example assuming 12 V.  You want the junction between the speaker and R3 to be about half way from either limit, so around 6 V.  To make 1.3 V from 6 V, the top resistor needs to be about 3.6x the bottom one.  36 k&Omega; for the top and 10 k&Omega; for the bottom should get you somewhere into a workable range.  Once you get things basically working, you can tweek the divider to get 1/2 the supply voltage on the top of the speaker.

To prevent whatever is happening on the other end of R1 from interfering with the DC bias of this section, put a DC blocking cap in series with it.  Let's say the lowest frequency you care about is 20 Hz, so 1 &micro;F will do nicely.