Hi, thank you for contacting JustAnswer.com. My name is Russell. I will do my best to provide the right answer to your question.
The power button blinking means either 'standby mode' or 'insufficient power from battery'.
Try connecting the Harmony One to the charger, then powering it up or bringing it out of standby there. The battery may be failing, and not providing enough power for it to come out of standby - all rechargeable batteries for electronic devices fail eventually, given enough usage. And if your Harmony One has been in use for several years, the batteries may have started failing.
Keep it on the charger for at least 6+ hours... that might overcome a slight polarization in the batteries. But if it still acts as you have described after that, replace the battery/batteries.