In your situation, your solicitor will probably sort it for you OK, because you've been maintaining this garden for over 12 years and so can claim adverse possession. It sounds like a Land Registry mistake that should have been rectified years ago - and certainly when you bought the property.
The statutory declaration is a simple legal document that your solicitor pulls off the LR website and it gets you to declare that you've been maintaining it for x years. Using it, the buyer will be able to get the garden registered as his own.
The indemnity insurance is belt-and-braces in case it goes wrong and someone emerges from the woodwork claiming the land - the insurance company would indemnify the new owner to buy back the garden.
The risk of this happening is tiny, so the policy premium is modest - perhaps £400 or thereabouts.