Kirch's comet was the great comet of 1680, not 1686.
It is used in Newton's Principia as Newton's first example of how to calculate
the orbit of a comet, to which he devotes many pages, showing how he used
a mixture of graphical methods and theory. There are problems with Newton's
calculation, which is very clever and cost him much thought; but I do not wish to go
into these difficulties in this context.
The comet was seen coming towards the sun, and later coming out, and there was
much discussion as to whether it was the same comet. Halley and Newton were
able to compute the orbit and settle the matter. Perhaps others could as well. It was
not an easy calculation, even for Newton.