You're confusing what the reference meant. If you use "that" with no comma, it will have approximately the same meaning as "which." This is correct, but will no longer be a complete sentence because you took the event of the sentence and turned it into a description of a noun. Here is that structure, but with more added to make it a complete sentence:
"After she completes the form which/that will be processed by Jason, it will be finalized."
"Processed by Jason" is no longer an event because it has become a description. Adding a new event makes the sentence complete. The form will be finalized.
If you keep your original sentence as is but use "it" and the comma, it will be correct.
"After she completes the form, it will be processed by Jason."
Here, the form is the subject of the sentence, and we say what is happening to it. It will be processed by Jason.
I hope that helped. I can try to clarify more if necessary.