120 v 2 speed mortor problem

The voltages you measure are fine. The fact that HIGH and LOW don't result in different rotational speeds is not, more on that later.

Your fan most likely uses a 2-speed (or multispeed) Permanent Split Capacitor (PSC) AC motor. There are two windings in there, the start winding that's connected in series with the capacitor, and the main winding that does the actual work. One end of the main winding is wired to the LOW lead, another one goes to the neutral (normally white) lead. The main winding of a two speed motor has one tap at a point in the middle of the winding, which is wired to the HIGH lead.

So, HIGH is actually using less turns of the winding hence lower resistance and higher current, more torque. Higher current in the main winding also makes the auxiliary winding (the one with capacitor) pitch in more, so to speak. Anyhow, the result is increased torque.

When LOW is connected and you are measuring voltage on HIGH, you are measuring it  at an intermediate point between 120V and 0V, so 80V on HIGH sounds just about right. Note the 40V difference.

When HIGH is connected, LOW is on the other end of the unused piece of winding and it has the induction forces generate additional voltage in it, which gets added to the 120V that's on HIGH. So, you add the same 40V to 120V and get your 160V - sounds just right.

The two speeds of this motor are actually two different torque outputs. Added torque does not necessarily result in higher rotational speed, at least not in fans where it's not easy to notice small differences without a stroboscope. It would not make much difference if the fan is too small a load even on LOW or too high a load even on HIGH. There may be mechanical reasons for the high load, too - bad or needing oil bearings of the motor or something else obstructing the fan's rotation. The fan blades loose on the motor shaft would make it a negligible load but there would be no air flowing, you would notice.  I have also seen large fans coupled to small motors and vise versa, which would result in high not being much, if any, faster than low.  

Anyhow, the voltages you measure are fine. If you want to investigate the speeds - oil the bearings, check the fan blades. Beyond that - not much can be done. If mechanical TLC does not change the speed, just run it on LOW all the time.

P.S. If this is a higher-end fan and it has a split phase AC motor with multiple windings, dual speed may actually mean two different speeds of rotation because it depends on how the windings are wired. In this (unlikely) case your wiring may be wrong. If you want to investigate it further, post the brand/model of the fan and better yet, the nameplate of the motor - I'll look it up. That's the only way to settle that for sure.