A family of Microsoft word processing software products for creating web, email, and print documents.
The failure to Keep With Next is due to the presence of a section break between the one-column and two-column sections (which is required because column count is a section property, not a paragraph property). This is true whenever there is a section break, regardless of whether the column count changes from one section to the next. Here's an example where both sections are one column:
Probably the best way to handle your text would be to use a two-column table with the borders turned off. It appears that all of the currently one-column text will fit into the left column of such a table, with the cells to their right remaining blank. In a table, Keep With Next will work.