Lithium ion batteries are pretty poorly recycled as is and so when I was looking through literature reviews I found only one mention of lithium ion polymer batteries being recycled. The paper that was using this method was also recycling lithium ion batteries the same way though so it wasn't clear if anything different had to be done.
The main thing that seems to be different with lithium ion polymer batteries is just the electrolyte. The electrolyte is often not recycled anyway since the main focus is typically on recycling the cathode because it contains cobalt and/or nickel and is profitable to recycle.
If you want a more detailed overview of the different processes lithium ion batteries are recycled this review is open access and explains the ways different companies currently recycle lithium ion batteries:
There's no mention of lithium polymer batteries in the review but this patent mentions recycling lithium ion and lithium polymer batteries together so I assume some of the same methods can be used:
There's no standard method for recycling lithium ion batteries but usually methods involve mechanical, pyrometallurgical and/or hydrometallurgical processing. Usually the batteries are first discharged and then shredded or dismantled.
If mechanical processing is carried out it usually relies on different physical properties in the materials that allow it to separate out components. For example, the electrodes are a fine powder so if you put the shredded battery through a sieve you can separate the electrodes from the other materials. The cathode and anode can also be separated using froth filtration since they have different hydrophobicities. This can allow the cathode to be isolated and further processed to obtain metals like cobalt if the battery has a lithium cobalt oxide cathode. The review goes into more detail about the other mechanical steps that can be carried out but there's quite a lot of them.
Pyrometallurgical processes involve putting the battery in a furnace and you end up with an alloy of metals (containing cobalt, nickel and iron usually) and a slag (containing aluminium, lithium, manganese and some other metals). The metals in the alloy can then be extracted.
Hydrometallurgical processes involve using solvents to leach metals. This is generally combined with other methods like mechanical processing so the battery components you have are slightly more homogenous and easier to process. Sulfuric acid is the most common acid used to leach cobalt from lithium cobalt oxide. The leached metals can be precipitated out and recovered.
It's a bit difficult to clearly explain how lithium ion batteries are recycled since it does seem like every company does it differently. This just mentions some of the processes that are being used at the moment but there's lots more as well that are being researched. I know this doesn't clearly mention how lithium polymer batteries are recycled but I hope it's of some help anyway!