Yes, so after 15 minutes of the door being open the interior light will timeout
Ok so I would avoid the recharging an SLA option personally, it's possible a few ways but there are better ways.
Depending on the current of your LED strip you could use the permanent voltage at the rear door light to replace the bulb.
The rear light unit AFAIK will have a permanent and switched feed as the rocker switch can be switched to on, unless it's different from the Traffic's I have played with.
If you use scotchlock connectors (not keen on them but they are best here) they could be removed in the future easily.
Failing that you may find an AUX voltage feed under the centre console or the radio will have a permanent supply, again these are all down to how much current the LED's are.
Alternatively there's solar, a small solar charger with some croc clips should be easy to rig up.
If you really want to go down the ciggy socket route and you are good with a soldering iron I would bash this together.
http://ludens.cl/Electron/12Vcharger/12Vcharger.html
Or you could buy an off the shelf 12v solar charge regulator or caravan split charger system.
They are two different things and wired differently but would do the job, either easily picked up from Maplin's, CPC or online.
I would consider this a PITA messy solution as the socket is used and wiring will be a pain.