Actually it was kind of simple in the end, though having it written out as a help would have been nice. There was an example on the page so I did an Inspect Element on it. The code is (for anyone else who stumbles on this thread):
<span onclick="toggleJpanel('#jPanel_ID#_of_jpanel_module');">Insert anchor text or image here</span>
In the onclick, look in your module list and fid the ID # of the module (mine happened to be 213) and replace ID#_of_jpanel_module with that #.
Unfortunately, if you use this method as opposed to the mouseover (which appears and hides automatically) then your cart stays open. So, I made a CustomHTML module, used EXACTLY the same code as I used to open it, but just changed the anchor text to "Close cart". When the cart is open you click that and it hides. The module appear in the slide-out window in the order they are in the module manager, so you can adjust whether your Close Cart is at the top or bottom.
This option works a treat. DO NOT use that Mini Cart option. Their support is almost non-existent and it does not work properly. I asked for support on 6/6/15 and still waiting. Asked for a refund 2-3 days ago and still waiting.