OP, you're still in the 'normal weight' range, albeit at the upper end. So in terms of framing this, it's not the worst thing.
However, I completely understand that's a significant weight gain for you in a short time, and you don't feel yourself.
For me, this is significant: im still exercising three times a week and run around after kids daily.
'Run around after the kids' makes no differences in terms of peri-menopausal weight gain. Moving, yes, but just activity like this will have no impact.
3 times week is good - but too little. I think you need to be exercising intensively 5 times a week, and ensuring it is varied exercise.
A PP mentioned weights - this is excellent as it works well for perimenopausal weight gain but will also help maintain muscle & avoid osteoporosis.
About exercise, I think it's so individual. I exercise 7 days a week - 5 of those are high-intensity, a mix of Spin, HIIT, weights, core plus 2 runs (one is an easy run so I don't wear myself out entirely). I always exercised a reasonable amount but for me, to manage a) my weight & b) my shape, more, and varied, exercise is important.
I know there are many theories about diet. I don't cut anything out but watch portions. I eat too much chocolate for sure but enjoy it & probably won't stop.
I still weigh about half a stone more in my 40s (I'm 46) than I did all my adult life (56kg, 5 ft 4). That's just about 9 stone & I work hard to make sure it doesn't increase. However, my physique is the best it has ever been, muscular, well-defined etc. my GP said impatiently, maybe that's the weight you're meant to be. I know my weight is fine, but that half stone seemed to come from no-where so I understand what you mean
I do think it's important to find what works for you re weight management. It gets so much harder to have excess weight & live a good quality of life when we are older.