Learn Russian: Я рассказал дочери такую грустную сказку, что она начала рыдать. - I told my daughter such a sad fairy tale that she began to sob.

We use a form of такой (such) because it is modifying a noun (сказку). It must match that noun in gender, number, and case, which is why it takes the feminine accusative singular form такую. If we were describing just an adjective without a noun (like 'so sad'), we would use the unchangeable adverb так.

Why is the word for daughter дочери and not something ending in an -е like typical feminine nouns?

Дочь (daughter) is a special feminine noun ending in a soft sign, meaning it follows the 3rd declension pattern. Because the speaker is telling the story 'to' her, the word is in the dative case. The dative form of дочь is дочери.

Why is рыдать (to sob) in the imperfective aspect, even though she completed the action of starting?

In Russian, verbs that express the beginning, continuation, or end of an action (often called 'phase verbs') must always be followed by an imperfective infinitive. Even though the starting itself was a single, completed event (начала), the sobbing is treated as an ongoing process (рыдать).

Is the comma before что required in this 'so ... that' structure?

Yes. In Russian, subordinate clauses (the secondary part of the sentence starting with что) must always be separated from the main clause by a comma. You cannot leave it out, even if you wouldn't pause there in English.