“What we must do is helping him”

Solution 1:

The sentence is grammatical, but not in the sense that you think.

A description of current events

Instruction:

"What must we do?"
"You must hold the door for him when he is carrying large packages and doesn't have a hand free to do so himself."

Result:

"What me must do ^{(hold the door)} is helping him."

Or to paraphrase with additional wording:

"The thing that we must do, and that we're doing, is actually helping him."
"What we were told to do in order to help him is actually helping him."

Or to simply replace the word:

"What we must do is benefiting him."

So, in this sense of the phrase:

✔ What we must do is helping him.

An instruction of future obligation

On the other hand, if what you're trying to express is not a description of how things are working out, but a directive about what must be done, then you need to rephrase it:

✔ What we must do is help him.

Solution 2:

It is not correct. The correct versions would be: What we must do is help him. What we must do is to help him. Although, personally, the first one sounds better to my ear. I believe that there is a typo in the workbook. Books are not infallible and do contain mistakes.