Three possibilities come to mind that you could check for. One is that the feed of power to the motor somehow does not work for the low-speed setting. This could be because:
(a) the switch is faulty and not sending power to the motor;
(b) the connections of the wiring from switch to motor are faulty, possibly with a dirty or corroded contact at a connection plug.
Both of those you could check for with a voltmeter. Use it to check the voltages into and out of the switch contacts when it is set to each of the two speeds. Then do the same at the motor to see what actually arrives there. At both connection points, carefully disconnect the wiring and examine the contacts for dirt and corrosion. Very mild oxidation sometimes can be cleaned off with a little scraping or even just unplugging and re-plugging a few times. Re-connect before going on to the next.
Another potential cause is a flaw of the motor. It could be that it has a separate low-speed winding that has failed. Or there MIGHT be a bad connection inside the motor case from the low-speed power line to the motor. Another version: the motor is so old that the bearings are worn, dry, or both. This means a lot more resistance to turning than a new motor, and supplying it with a normal reduced voltage for Low Speed is not enough to actually start up the stiff motor. IF you can, try just adding a small amount of oil to the exposed motor output shaft where it comes out of the bearing, then running it on high for a little while to work in the oil.See if that makes any difference.
Third possibility is similar. Maybe the stiff dry bearings are not IN the motor, but part of the metal linkage arms between the motor and the actual wiper shafts. ALSO consider the little shafts themselves that the wiper arms are attached to. They go though some mounting sleeves that act as bearings. Try some oil on them as well as on any mechanical connection points in the metal wiper linkage parts.
Among all of these, the more likely (for a system that does work partially) and surely the easiest to try first is the oil on all moving parts thing.