Hey @moquo, sorry about that. The backup feature isn’t part of the free plugin. It’s included with the paid Jetpack plans (Security, Complete, or the standalone VaultPress Backup plan). You can see what’s included at https://jetpack.com/pricing/.
As for your design breaking after uninstalling, I’d like to help you get that sorted out. Can you share your site’s URL and describe what looks different? For example, are images missing, is the layout shifted, or is the styling gone? That’ll help me figure out what happened and point you in the right direction.
You can also reach us directly at https://jetpack.com/contact-support/ if you’d prefer hands-on help. Either way, Jetpack doesn’t change your theme files, so whatever’s going on should be fixable.