Your plan to install on a desktop machine and then move the drive back to the laptop is impossible. When the Install is done, the process will analyze all the hardware devices in the desktop system it is running on - and there are probably at least 20 of them in the mobo, plus attached devices - and install all their drivers. THEN you move it to a totally new system that has an entirely different set of components. The very best you could hope for is that it might boot with lots of problems. The more likely scenario, even IF it could work, is that you could instead do a "Repair Install" on the Laptop machine. That is, you could boot from the original Install CD and choose Repair Install, which completely re-analyzes all the drives and fixes them up. But WAIT!! That requires having an optical drive in the laptop to do this, and you say you can't do that! In fact, if you had one, you would just do the proper Install on the laptop to begin with.
It appears whatever you do will cost $$. I suspect buying an exterrnal optical drive is cheaper than having the internal one replaced. For a faint hope - what about trying to simply clean the optical drive, just in case its only problem is dirt?