DIAGNOSIS The speaker isn't getting power. This is almost always a power supply issue, not the speaker itself being dead.
FIX Unplug the power cable from the wall and the Nest Mini. Use the cable from a phone charger or another Nest Mini to test. Plug that known-good cable directly into the Nest Mini and then into the wall. If it powers on, your original cable is faulty. Google sells replacements.
IF THAT DIDN'T WORK Perform a hard reset. With the speaker powered (using the test cable), press and hold the center of the device (where the lights are) for about 10 seconds until you hear a sound confirmation. This reboots the system and can clear a software hang that prevents startup.
HEADS UP The Nest Mini 2nd Gen's power cable is permanently attached; you can't replace just the USB-C end. If the cable is damaged near the speaker, the entire unit needs replacement.