The two sentences just before the ones you quoted indicates that at least two (the plural is used) and likely more ("surrounded") Balrogs were dispatched to stop <PERSON>: This gives us a lower limit.
Thus it was that he (<PERSON>) drew far ahead of the van of his host; and seeing this the servants of Morgoth turned to bay, and there issued from <PERSON> to aid them. There upon the confines of Dor Daedeloth, the land of Morgoth, <PERSON> was surrounded, with few friends about him. The <PERSON>
The main problem lies in determining the upper limit, since the total population of Balrogs varied a lot during the redaction of what would later become the Silmarillion, from hundreds to at most seven:
There came wolves, and wolfriders, and there came Balrogs a thousand, and there came worms and drakes, and <PERSON>, Father of Dragons. War of the Jewels, The Grey Annals - Year 472 ~230
The idea that <PERSON> disposed of a 'host' of Balrogs endured long, but in a late note my father said that only very few ever existed - 'at most seven'. Book of Lost Tales, Tome 2, The Fall of Gondolin
The upper limit on the number of Balrogs fought by <PERSON> (including Gothmog) is therefore "three up to a thousand" (but likely no more than a dozen top, see the following rational)
Let's focus on the number of Balrogs he defeated, next. 466
I would say "none", since no kill is mentioned in text, and the two Balrogs defeated by <PERSON> and <PERSON> respectively are seen as big deals in the rest of the published Silmarillion, which aims at a certain coherency (see the book forewords).
In fact, it appears that <PERSON> obeys the rule of ConservationOfNinjutsu. Depending of their total population at the time of redaction, a single <PERSON> could be considered narratively as anything ranging from EliteMook to PhysicalGod:
The early conception of <PERSON> makes them less terrible, and certainly more destructible, than they afterwards became: they existed in 'hundreds' (p. 170), and were slain by <PERSON> and the Gondothlim in large numbers: thus five fell before <PERSON>'s great axe Dramborleg, three before <PERSON>'s sword, and two score were slain by the warrior's of the king's house. Book of Lost Tales, Tome 2, The Fall of Gondolin
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