The main answer might be to use the Pen Tool to redraw; the Pen Tool is the main tool, in many/most cases leading to both quicker and better results.
You may obtain some simplification with Object>Path>Simplify.
The Simplify command seldom works unless it is a very complex path with many small segments.
However besides the fact that the pen tool might work if you get good at it and you kind of have to get good at it. You can use the convert anchor point to make sharp edges you can also us it to break the connection between the two handles of any anchor point so one can be straight and the other curved by click the tool on the handles end point and dragging instead of clicking on the anchor point itself.
This can be helpful for what you want to do.
Thank you both. I think I'll have to redraw most of it.
For my part you are welcome, Shenzi.
You may consider a combination of drawing with the Pen Tool and joining paths, including parts of basic shapes such as squares/rectangles, circles/ellipses, polygons, etc. That is one way to get segments that are parts of circles, right angles, etc.
In this connexion, you may cut the paths at Anchor Points with the Scissors Tool, and you may Object>Path>Add Anchor Points to get the ones you need.
You may join separate paths by selecting coininciding end points with the Direct Selection Tool and Ctrl+J/Cmd+J.
With a cut/open path, you may continue drawing, clicking/clickdragging the end point with the Pen Tool.
You may also use the Shift key while using the Pen Tool to get the desired directions/angles.
And many other options.