The modern meaning of "cure" is different to the German, but in the past it was used more like the German, hence my suspicion that "sweating cure" might have been used at some time in the past (e.g. until the early 1900s).
6 a A particular method or course of medical treatment. M19.
b A period of residence at a health-resort, under medical regimen, in order to restore or benefit one’s health. Cf. KURSAAL. L19.
b W. S. MAUGHAM The cure at Montecatini had not done him the good he expected. (1944)
AFAIK this use came straight from the German Kursaal
. In the 1800s and early 1900s there were Kursaal
spa resorts across Europe - there is still one in Southend (UK), for instance. http://kick-butt.co.uk/kursaal/
Oh, and your other point.
Find me a dictionary that doesn't translate "das Schwitzen" as "sweating". How else would you translate it? Here's an example sentence using "das Schwitzen":Ein viel besseres Mittel zur Temperaturregulierung ist das Schwitzen.http://www.zeit.de/2002/32/200232_stimmts_hei...
How would you translate that?