Diagnosis
The mode button itself is likely gummed up with dried toothpaste, mouthwash, or water minerals underneath its silicone seal, preventing a proper electrical contact.
How to Fix It
We need to clean under the button. 1) Unplug the charger and remove the handle from the base. 2) Rinse the handle under warm running water, focusing the stream directly on the mode button. While rinsing, rapidly press the button dozens of times to work water in and out and dissolve any gunk. 3) Shake out excess water. 4) Now, with the handle OFF, press and hold the mode button firmly for 30 seconds. Sometimes this helps re-seat the contacts. 5) Let it air dry completely for a few hours before trying to use it again.
If That Didn't Work
The internal membrane switch behind the button might have failed. Try a factory reset: With the brush off, hold the Power and Mode buttons together for 10 seconds until the lights flash. If it's still stuck after that, it's an internal electronics fault.
Heads Up
After every use, wipe down the entire handle, especially the button area, with a dry cloth. Preventative drying is the best way to keep those sealed buttons working long-term.