Forget about arguing with the manufacturer. It's the retailer who is responsible for putting matters right.
From the point at which you make a purchase up until the point where you are legally deemed to have 'accepted' the goods you have the right to obtain a full refund if the goods are faulty. As you've never managed to get the mouse to function properly, it's likely that a court would say that you've not 'accepted' the goods (in the legal sense).
Even if you reach the point where the law deems that you've accepted the goods, you still have rights under consumer protection laws. Although your right to a refund disappears upon 'acceptance' of the goods, you still have (for 6 months from the date of purchase) the right to have a faulty product either repaired or (more likely) replaced by the retailer. During this 6 month period there is an automatic legal assumption that any fault which develops was 'inherent' (i.e. present at the time of manufacture). So, you don't need to prove that you followed the charging instructions, the retailer is obliged to assume that the fault lies with the product.
For the sake of completeness (although it seems unlikely that it's relevant here), I'll add that your right have a faulty item repaired or replaced by the retailer continues for a total period of 6 years after purchase but, once the initial 6 months have passed, you might be required to prove that the fault was present when you purchased the item.
Just take the mouse & keyboard set back to the retailer and insist that the matter is rectified.
Chris
PS: From a technical viewpoint, I'll add that all rechargeable batteries need to be charged and fully discharged several times before they'll hold the maximum charge but it sounds as if you've already done this so it definitely seems as if you need to contact the retailer.