extract vbios from ASUS n55sl

[pfdd98]

MXM is simply the format/type of card connector it uses.  And yes, it has it's own vBIOS, may or may not be same one inside BIOS rom (Usually is)
You'd have to remove the card and dump it physically if you wanted to be 100% sure what is on there.

If I were to update the vBIOS inside the BIOS yes, it would need to match what you see above since that is what was taken out of the BIOS.
What does your card look like in GPU-z?  Show an image, be sure it's showing the Nvidia info, if you see Intel info, switch it at the bottom drop out menu
https://www.techpowerup.com/gpuz/

Oh ok, so since I'm crazy and I don't love money, I dumped the n55sf bios to see what I will get , sourced from asus website. It's working perfectly and now on gpu-z shows the 555m with the same clocks.

[Lost_N_BIOS]

What did original card BIOS show up as in GPU-z

[pfdd98]

What did original card BIOS show up as in GPU-z

Nvdia gt 635m, everything is the same except the name

[Lost_N_BIOS]

So what did you do, swap CMSCore regions, or cut/paste both vBIOS from other board? I'd put back in the original ones, however that wouldn't make sense, since original is 555m too, I don't see how it could be 635M