These are very similar constructs.
If you reformulate these sentences into questions in English:
How carried away did he get with the cleaning? So that he lost track of time.
How many times did I try to reach out to her? So many times that I lost count.
, you can notice that these questions allow an answer with a quantifier: "much", "little", "more than you" etc.
English has one generic interrogative for questions like these, namely "how": "how much", "how long" etc.
Russian, however, has at least one more: сколько, and the corresponding determiner столько. In English, it's just a variety of "how" ("how much" or "how many"), with corresponding "so" ("so much" or "so many").
This way you can probably see that both these constructs are in fact varieties of the generic "so ... that", except in Russian "so much" has its own word (столько) while others are just the generic "so" (так).
So столько раз, что just means "so many times that", and is not at all different from other "so ... that" constructs which are conveyed by так ... что.
In fact you can even rephrase the original Russian sentence this way:
Я так много раз пытался связаться с ней, что потерял счет.
which makes it parallel the other one even better.