When the Powerbutton on your JBL Go personal speaker is not working, this likely means one or more things as follows.
Dead Battery. The internal battery is depleted and needs a recharge for continued speaker operation. Use a commercially available USB charger to do this.
Faulty AC Adapter. The power adapter charger is defective. So replace it with a charger that delivers 5 volts, at 0.5 amps (2.5 watts).
Faulty DC Input Port. The micro USB input port for charging the speaker may be cracked, corroded, bent, or otherwise damaged. Indeed this is a typical problem we see on this speaker. So replace this if you have the skills.
Spoiled Battery. That battery no longer recharges due to advanced age, repeated overcharging, very long storage without recharging, storage in temperature extremes, or heavy use. Replace the battery in this case with one that delivers 3.7 volts at 600 milliamp hours (2.22 watt-hours).
Contaminated Button Panel. The button panel has dirt, sand, or other debris preventing the Powerbutton from making proper contact when you press it. For this, you can, if so skilled, clean the switches by taking apart the speaker, and blowing, brushing out, and polishing the contact area.
The Button Panel Has Failed. The switch panel is defective, and the solution here is to replace it. These parts are often available at eBay.
Faulty Speaker. Your speaker is broken beyond repair. If you decide that this is the case, then the best bet is just to get a new speaker.
An Echo Dot 4 clock speaker beside a JBL Go 1 BT speaker