How to get an advisor to finish their part of a publication when they have not done anything on it for over a year?
Is a year and a half a normal amount of time to write on a paper in which the research is finished? 


No, this paper should've been finished a year ago. 

The research is done, so you are not waiting for something else to fall into place for your paper. You already tried writing your former advisor's portion of the manuscript to no avail. Your former advisor did not even catch a critical flaw in your thesis. I hate to say it, but it sounds like your former advisor is quite a bozo.


  How do I politely convince her to hurry up?


If you've not already done so, send your former advisor your "final" draft of the manuscript (including the parts you wrote for her). Give her a timeline for returning suggestions/modifications to you. If that doesn't work, you might have to get on the phone and talk to her about how important it is for you to submit this paper. If this fails, you may need to move on, and hope that at some point in the next year or two that she finally matures enough to help you out.