How to build a programmable LED strip?
Your best bet is to find a single color, high power, leds strip and command it using a uC (ex: arduino). I try to explain why.

The name WS2812 (and alike) refer to the chip inside the LED, not to the LED the chip is embedded in. In some early versions of those strips, the chips aren't embedded and you can see them on the strip. In theory you could buy those chips (or any other 1-wire communication chip) and add the LED yourself but the job is pretty nasty.

Going a bit more in detail. To power the LED you need a transistor, and the command the transistor you need a command channel. This channel can be 1-wire (ex: WS2812), Two Wires Interface (aka I2C), or more wires (example: SPI). Or you can make a complex matrix made of rows and columns, but then you might need to introduce extra diodes (ie: extra wiring complexity, extra soldering work) to avoid things like "ghosting". 

Another chance is to connect multiple arduinos on the same I2C/SPI bus and use each GPIO for exactly 1 led only; you don't really need the whole arduino, you can use a bare minimal circuit to turn on a big AVR chip, or any other chip with many GPIOs. The atmega2560 have 60+ GPIOs (if I remember correctly); this means that you could individually address 60+ LEDs each atmega2560. So you could embed 1 atmega1280 (or 2560) per each one of your 25-leds rows, directly on your pcb.

But still, have you an idea of how much wiring complexity, and how much soldering work you'd have to do? If you can find an high power, single color, individually addressable leds, strip ... then connecting the strip (or mutiple strips) to an arduino is trivial. On my arduino nano v3 I'm controlling more than 300 (individually addressable) leds using 3 pins. But they are WS2813-clone strips (ie: RGB, and not high power).

In general, I'd leave the LEDS part to the machines, and configure the machined strips in the configuration I need. Just glue the strips on your pcb! :)

There's another problem then: heat. High power leds are soldered on aluminum boards because aluminum works as heatsink, or contact surface for a big heatsink. You can't find those high power leds mounted on flex-pcb because they would burn after 2 minutes of usage. So, if you are using high power leds, be sure to make your pcb in the proper way (ie: leave big, BIG, copper areas for dissipation purposes; and probably use those areas for connecting a bigger heatsink).