Where does Rashi say [if at all] the shibud Mitsrayim was for 113 years [of which the last 86 were the קושי השעבוד]?
Levi died at 137 (Shemos 6:16). Rashi ad. loc. says that the figure is given just so that we know how long the Bnei Yisrael were enslaved in Mitzraim, as the enslavement didn’t start until Levi, the last of the brothers, died.
But he doesn’t explain.
Mizrachi there says that Levi was in Mitzraim for 94 years, and 210-94=116 years. He doesn’t explain where that number comes from.
Sifsei Chachamim quotes a tradition that all of the brothers were born within 6 years of each other, besides for Binyamin. Levi had to have been in the third year, as Reuven and Shimon were born before him. Thus, Levi was 4 years older than Yosef.
As Yosef was 30 when he became viceroy (Bereishis 41:46), Levi must have been 34 at the time. Yosef became viceroy at the beginning of the 7 years of famine (41:32 indicates that it will start immediately - see Rashi there), and his family came down in the second year of famine (45:6). After these 9 years, Levi was 43. Subtract 43 from 137 to get 94. As above, subtract 94 from 210 to get 116.
As for the 86, Rashi to Shir HaShirim 2:11 writes:
כלומר שמנים ושש שנה של קושי השעבוד נגזרו עליכם והלכו להם משנולדה מרים הקשו המצריים שעבוד על ישראל ולכך נקראת מרים על שמררום
That is to say: 86 years of harsh labor was decreed upon you and they went for them (?); when Miriam was born, the Mitzriyim increased the labor, and therefore she is called Miriam, on account of their embittering (shemarerum).
Credit to Yaacov Deane for drawing my attention to this Rashi.
The number 86 comes from the fact that Miriam was 3 years older than Aharon (Tosfos to Bechoros 4a, from Shemos Rabbah 1:13); as Aharon was 83 upon leaving Mitzraim (Shemos 7:7), Miriam must have been 86.
How about Rashi to Shir HaShirim 2:11 which says:
הגשם. שהוא טרחותו של סתיו חלף והלך לו כלומר שמנים ושש שנה של קושי השעבוד נגזרו עליכם והלכו להם משנולדה מרים הקשו המצריים שעבוד על ישראל ולכך נקראת מרים על שמררום:
Rashi to Shemot 12:40 says explicitly that the actual servitude in Mitzrayim was 210 years.
שלשים שנה וארבע מאות שנה FOUR HUNDRED AND THIRTY YEARS — Altogether from the birth of Isaac until now were 400 years, and we must reckon from that event, for only from the time when Abraham had offspring from Sarah could the prophecy (Genesis 15:13) “Thy offspring shall be a stranger” be fulfilled; and there had been 30 years since that decree made at “the convenant between the parts” until the birth of Isaac.
It is impossible to say that this means that they were 430 years in the land of Egypt alone, for Kohath was one of those who came into Egypt with Jacob (Genesis 46:11); go and reckon all his years and all the years of Amram his son and the whole eighty years of Moses, the latter’s son, until the Exodus and you will not find that they total to so many; and you must admit that Kohath had already lived many years before he went down to Egypt, and that many of Amram’s years are included in the years of his father Kohath, and that many of the 80 years of Moses are included in the years of his father Amram, so that you see that you will not find 400 years from the time of Israel’s coming into Egypt until the Exodus.
You are compelled to admit, even though unwillingly, that the other settlements which the patriarchs made in lands other than Egypt come also under the name of “sojourning as a stranger” (גרות), including also that at Hebron, even though it was in Canaan itself, because it is said, (Genesis 35:27) “[Hebron] where Abraham and Isaac sojourned”, and it says, (Exodus 6:4) “[the land Canaan], the land of their sojournings wherein they sojourned”.
Consequently you must necessarily say that the prophecy, “thy offspring shall be strangers… [four hundred years]” began only from the time when he had offspring. And only if you reckon the 400 years from the birth of Isaac will you find that from the time they came into Egypt until the time they left it, was 210 years (as alluded to in Genesis 15:13).
This was one of the passages which they altered for king Ptolemy (Mekhilta d'Rabbi Yishmael 12:40; Megillah 9a).
The first 124 (210 less the 86 mentioned from Shir HaShirim) were not as harsh. After the last of the sons of Yaacov was niftar, things started to change for the worse and it was for this reason that Miriam was given her name. Miriam has a connotation of bitterness (מר). It is also worth noting that 86, the number of years that the bondage was bitter, is the gematria for אלהים which is the name of G-d associated with the quality of severity/judgement (גבורה).