Covers How to Fix Sony SRS XB13 Power Button Not Working here. Now when this button stop working, this likely means one or more things as follows.
How to Fix Sony SRS XB13 Power Button Not Working: The Reasons
Dead Battery. The internal battery is empty needs a recharge for continued speaker operation. Use a widely 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 Power Port. The micro USB input port for charging the speaker may be bent, corroded, or otherwise damaged, an often-seen on this model. So replace this if you have the necessary skills.
Bad Battery. That battery no longer recharges due to extreme age, repeatedly overcharging, very long storage, or heavy use. Replace the battery in this case with one that delivers 3.7 volts at 1,400 milliamp hours (5.18 watt-hours).
Dirty Button Panel. The button panel has sand, dirt, or other debris preventing the Powerbutton from making decent contact when you press it. For this, you can, if so skilled, clean the switches by disassembling the speaker, and blowing or brushing out the contact area.
Button Panel Has Failed. The switch panel is defective, and the solution here is to replace it. These parts are often available on eBay.
Faulty Speaker. Your speaker is broken beyond repair. If you determine this, then the best bet is just to get a new speaker.