If it's not a typo, maybe it's a kind of "anacolute" or something like that: the author begins writing the sentence using a tense, in this case past tense, and then, in the same sentence, changes the tense, using a present (the present can be used with a past meaning when telling stories). However, if that were the case, it would be a rare one, anacolutes are usually different, they are represented by the interruption of a certain construction in a sentence and its substitution with another construction.

It's weird - I don't think it's wrong, but it's a very odd choice. The words that the author wants to make parallel are "jump" and "begin." In other words the writer is saying: "She than started to - - jump on the bed - begin singing ...." Usually, a writer would include the entire infinitive both times: She then started to jump on the bed and *to* begin singing "Five little monkeys jumping on the bed." I'm not 100% sure if you need that second "to," but it is odd (and slightly confusing) not to put it there. In your version, the two words that are parallel are "started" and "began" or in other words: "She then - -started to jump on the bed -began singing ...." And yes, "began" would be the correct way of writing that sentence (which is slightly, but not substantively different) than the author's sentence. Final comment, the punctuation leads me to believe it's a British book - it may be a UK English peculiarity.

Hi Jeff, excuse me, what do you mean "a guy corrected my sentence"? So maybe the sentence is wrong? Is that guy a reliable source?

You are right, 'began singing' is correct. However, with no context, it is difficult to be sure, because the author might be reporting colloquial speech for some reason...it's an odd sentence.