Which jQ menu plugin to choose to make it easy to implement the presented look and feel?

Question

JavaScript

Good day, friends.
I worked with PHP for a friend, we worked well together, after that he asked me to remake the menu for him on the site. The site itself is quite amateur, has been successfully living for more than ten years. Side menu, implemented on frames. By default, there is only one frame, everything is dumped in it, when a specific book is opened to the right of the main menu frame, a second frame with a list of chapters sticks to it.
It all looks like this in 2015:

Nevertheless, the comrade really asks not to change the appearance from the word at all, but to do the same on a more modern platform so that when you refresh the page, you don’t have to scroll the long menu frame again (and look for where you just were) and that the address bar should have a specific path to a specific open page, not index.htm, no matter what we clicked.
Well, I personally jars quite the presence of as many as two scroll skids.

Accordingly, I am looking for a jQ menu plugin or some other good solution that will allow, if necessary, to "stick" another one to the static side menu (without wild whistling animations) and, preferably, has the function of replacing the scrollbar with a "down" button, or at least a neat scrollbar.