The display itself is probably still intact, but there are a few things you should check before diving into the electronics. Try completely shutting off the car radio - unplug the power connection for about a minute and then plug it back in. That resets the device and can sometimes fix a black-screen problem. When you turn it back on and still see nothing, look for the contrast dial or brightness buttons - they're often pretty hidden on Kenwoods, but if the brightness is set to 0, you won't see anything on the display even though it's working.
An important thing people often miss: check whether the power cable is really sitting properly in the ISO socket. On older car radios, the contacts tend to loosen up from vibrations pretty easily, and then the display gets minimal power (that's why the music still plays), but not enough to turn on the LCD. Try pushing down firmly on it or unplug it and plug it back in.
If none of that helps and the display really is broken, I'll be honest with you: repairing it yourself is a pain. On this model, the display is basically glued in, and if you start trying to get to the circuit board, you'll need special tools and a lot of patience. A professional shop can do it way faster, and for an older radio like yours, a replacement display won't be outrageously expensive afterwards.