Laptop screen cracked, unsure about replacement

Laptop screen is badly cracked, needs replacement.
Original laptop is a Compaq Presario CQ61 with a 1 CCFl screen (n156b3-l02 rev.c2). This is with a 30 pin connection and the pink/white backlight cable.
I already have an LED Screen from a dead laptop (LP156WH4 (TL) (N2)). This is with a 40 pin connection and no other external cables.

As it stands right now I have the cable going from inside the laptop proper, splitting in 2, one fork going into the left side of the inverter and the other turning into a flat cable and going into the screen as a 30pin connection. There is also the pink/white backlight cable going into the right side of the inverter.

I'm wondering there is any way to use the LED screen using a 30pin to 40pin converter or possibly to purchase a 40 pin cable outright and ignoring the inverter competely or something along those lines. I simply haven't been able to figure out to what level the laptop is built around using that specific screen.

Thanks in advance.