Must she make up the fast for two days during which the questioner saw a very thin streak of blood that did not stain her clothes, but rather she saw it when she checked for the onset of menstruation, and then menstruation began after those two days?

Waswasa (obsessive whispering) is a serious illness, and to get rid of it, one must ignore it and pay no attention to it. Menstruation is only confirmed by seeing blood on the outer part of the vulva, not within its inner part. If you are certain that a thin string (not waswasa) has exited to the outer part of the vulva, you must make up for those two days. However, if you are unsure about its exit, the default is that it did not exit, and you are not required to make up for the days.