You remove the baseboard, cut through through the old floor using a special kick space saw (they an go flush to a wall), tear it out, and then you can go up to the wall with the new, and the ends will be concealed by the baseboard.

Does this link below help? We had a pony wall removed between the kitchen and dining area but I didn't watch what procedure the guy doing the work did, except he didn't have to take all the flooring up to fill in the area left empty from the pony wall removal (about the same amount of space as you need to fill). Here is a link that might be useful: Repairing laminate flooring

It is the light color that is difficult to match. As geoffrey_b said it is difficult to match due to age. I agree that it will age with time. I too have an older home (66 years) and chose to have the old floors refinished so the difference is not so great. I attempted to duplicate the stain by refinishing my daughter's room by myself. I stripped, sanded, and stained the floors and they are practically indistinguishable from the old floors. Of course I have not been able to walk without pain for weeks after trying to dance around with the sander I rented, but it is all good. I agree with rwiegand that you might try a sealer in a slightly darker color in an inconspicuous spot and see whether you can match it. Sealers are made in all colors and I have actually mixed colors in order to match when refinishing furniture.

I suspect I will have a similar situation if /when I replace the vanity in my powder room. I think in your case if you could just put white tile ( similar in size to what you have) in that space, that might work. You won't really see that it doesn't have the pattern in the rest of the floor, and will just look like it had the continuation. If tiling is too complicated then I guess enduring's suggestion would be easier. Not sure about matching and all, but then again, since it will be under the vanity I guess there's a lot of leeway with shading and all.