YTA - And to break it down for you:

My wife was not fan of this and wanted my sister out and told me to give her our other accommodation which we own to my sister whilst she settles for divorce but my sister wants to live with us as she gets lonely and won’t have anyone cook for her because this whole thing is stressing her out.

First off, you go against your wife's wishes and move your sister in despite already offering her your other home, purely on the basis that she wants around the clock entertainment and a personal chef.

My wife has left our business cus of broken leg so she can help with the cooking for my sister.

You've never had a broken leg before have you? So I'll break it down, whilst technically both arms are operational enough to be able to slave cook, have you considered that most of your wife's strength is most likely going towards using her arms in crutches to be able to get around? This must have been extremely tiring for her, and she was probably in quite a lot of pain whilst your sister sat on her arse moping.

Since the start of the year my wife been wanting Dyson Air thing for Hair(I don’t get the excitement)

It's not about you, it's about your wife, and what she's gets excited about

Because there was no option for my sister I just removed the name sticker and put my sisters name gifted the Dyson I bought for my wife to my sister. Keep this in mind I bought my wife extra gucci perfume she STILL got the gift.

I can't explain the level of asshole here, imagine you really wanting something from your wife, something that'd put the biggest smile on your face on Christmas day, then watching your wife's sibling open that exact present from her whilst you got some shitty aftershave, I guess it's fine if she makes up for it by getting 2?

Until my mum told me that she saw my wife’s face during the gift giving and my wife looked hurt apparently

Apparently? Wake up idiot, she WAS and IS hurt. Any idiot and their dog would realise she was hurt. The fact that you didn't actually makes you a crap husband despite all of the above.