You are already on iOS 5.x so the iPad does not need a laptop for a basic restore. You will need internet via wi-fi.
So, laptop or not, you should be able to restore the iPad to basic function. Restoring your data will go faster if you operate off a local backup off the laptop, but if you do a backup via iCloud you can restore from there as well, but it takes a fast internet connection to work well, especially if you have a lot of apps.
That said, you may not want to restore from a backup. This can easily re-introduce whatever caused the problem in the fist place.
In either case, be absolutely to save off everything important before you do before any kind of restore. Go through all your apps, Camera Roll, etc. Anything that is not an iTunes purchase needs to find a safe home off the iPad. With the exception of some movies and TV Shows, almost all iTunes content can be downloaded again. That's not true for any documents or data you've stored in apps.