RE the Land Registry - if a interest in property is registered, then you can get what is called an 'official copy' of the title register extract from them (which is what I presume you mean by Land Registry 'copy'). There is no original of this as this is a central register - the downloaded copy is sufficient to show title from a legal perspective.
However, it is not clear to me if you are asking about your SIL's title to the property (which is presumably registered and freehold) or the lease to your mother. If the lease to your mother has been entered into formally and is longer than 7 years, then that too should be registered at the Land Registry and you can get an office copy of that title too. In that instance there would be two titles, the freehold one and the leasehold one, both of which official copies would be available for.
In addition to that is the lease document itself (if formally entered into). If that is registered, then you should be able to also request a copy of that from the Land Registry if you cannot find the original. That may or may not suffice for your purposes, but I would have thought it would suffice if submitted along with the official copy registers showing the relevant ownerships.
If the lease is not registered or is too short to be, then you are going to need to original or a certified copy of it to show entitlement.
I'm not completely clear as to what the plan is however, are you intending to enter into another lease with the mother, or share occupation with her under her lease? You'd need to make sure that her lease allowed that under its terms if so.
If its an informal arrangement, then pps have made suggestions in relation to that.
Hope that helps.