Well, when you first pick up a functioning mouse of this type it does “blink” from the bright red LED to a dim red light… once. And then stays dim until placed back down on surface. But, no continuous blinking. (Not using it on a glass surface?)
Check the cord at the point where it exits the main mouse housing as that is the typical stress point where these mice fail most often (i.e. stretched, broken or strained wires inside that little cord). If you’ve ever tried to re-wire or correct this broken wire strand condition you’ll understand why most simply just get another mouse.
Looks like yours is “twisted a bit” (have seen worse) where it enters the USB plug at the keyboard. Lots of miles and cable twists on these olde mice. Carefully try “un-twisting” any kinks there and where the cable exits the mouse… to see if it possibly changes. If not, well then >>> new or newer mouse.
Then there’s always the reboot & rebuild desktop / zap the PRAM ritual voodoo - before complete surrender.
*You might also try plugging the mouse directly into a USB port on computer back - instead of into the keyboard.