You might need separate voltage regulators for the digital & analog circuits... Depending on what kind of power supply you have, that might require a new/different power supply. 
First, I'd just try a capacitor (1000uF or more) across the audio power supply. A diode in series before the capacitor usually helps too. The diode prevents the digital circuitry from discharging the capacitor... The capacitor can only discharge into the audio circuit. (The diode will lower the supply voltage by about 0.7V, so hopefully this isn't a problem for the SpeakJet chip.) Another diode & capacitor on the digital/LED side may help too.