Manual transmission cars have two types of linkage, mechanical or hydraulic. I'm not familiar what the Lexus uses, but if it's the hydraulic type the slipping would not be corrected by adding fluid to this part of the clutch mechanism. Pushing down the clutch pedal, whether it's hydraulic or mechanical just serves to disengage the clutch and does not have anything to do with the slipping symptom that you described. The pressure plate or the clutch plate or both are probably worn out. The clutch plate has a friction material similar to brake linings and when that is worn down slipping occurs.
I don't know how much they charge for this type of repair, but you can get an idea by calling a few repair shops. Naturally, Lexus Dealers would probably charge the most.