Not really possible to answer (other than emotionally) without more info, but I get that you want perspectives while you mull it over!
How are you considering paying it off? By increasing your payments to £700ish a month and paying it by this time next year? Would that leave you miserable this year?
Or you have that 13k in the bank and are wondering what to do with it? Or something else?
What else might you do with the 13k? (if you have it). I see you have emergency savings, which is cool. Are your pensions on track for the retirement you want? If not, those take priority, the sooner the better. Are the kids educational funds enough, or could they benefit from being bolstered? If the kids are little, I would temporarily prioritise the mortgage. If they are late high school, I would not. Do you have any plans to move or do work on your house?
I’m assuming you have no other debt, from what you’ve said (otherwise that’s first to be paid). I had a finance agreement and no mortgage and then swapped them and it doesn’t feel any different to me (but then I’m very comfortable with a mortgage. YMMV) It makes no practical difference, as if you ended up in court defending a repossession, for any reason they would take literally anything else before your home.
I would increase the payment, if you can afford to, but I (probably) wouldn’t use savings to pay it. Interest rates and inflation are going up, so it makes sense to pay it down sooner if it’s on a variable rate. Or you could fix. I don’t think I would do that though.
If it were me, and I had emergency savings covered and if long term savings (kids’ ed fund and pensions) were on track for their goals (I bet they are. You sound very on top of it). I would have a lean year and pay it off ASAP. If it took more than a year, or doing that would make me too miserable or affect the kids too much, I would increase the payments to pay it in 3 years or 5. Whatever is doable. (I’m assuming your income is such that you can do this and still be somewhat comfortable, but I don’t know!). I wouldn’t leave it as a loose arrangement.
If I had the 13k in the bank, I would look at what interest I was getting. If I was getting more interest (fairly unlikely), I wouldn’t use it. I’d stop saving into it and increase my payments, until the mortgage rate was higher than the savings rate and then I’d pay it off. (But I wouldn’t have those savings in cash in the first place anyway). I’d probably look at spending half the savings on solar power and a battery (if suitable) and put the savings on energy into the mortgage. I would also look at a secondhand EV for day to day use and put the fuel/tax and insurance savings towards increasing the mortgage payment. This would save me personally about half the £700ish mortgage payment to pay it off in a year, and I’d increase my payments and go without a little bit (but not loads) to do that. I’d happily go without a holiday (or downgrade it) and cut down (but not completely out) on beauty, eating out, treats or a car or delay home improvements. I wouldn’t cancel my kids activities, but I’d be prepared to delay starting new ones, for example. After that, I’d drip feed the cost savings into pensions, kids’ funds and investments equally, unless the first two were already as good as they need to be. Then I’d just invest it.
I might also be prepared to dip into emergency savings a bit. If you have (say) six months emergency savings for essential bills, the mortgage would be part of that, so you won’t need so much in reserve if the mortgage is paid. It might pay to use some to reduce the obligation earlier. Don’t wipe them out, of course. But maybe reduce them a little.
There’s a lot of assumptions here. Maybe you have a north facing roof, or don’t use a car. Maybe your kids are babies, or maybe they’re off to uni next year. Or maybe they’re babies and you plan to educate privately. Maybe you lie awake at night worrying about the mortgage. Or the interest rate. Maybe you would be utterly miserable without monthly weekends away from a stressful job. Maybe your job is very secure, but stressful. Maybe it’s already looking rocky. I can’t know. There’s nothing wrong with any of those things, it’s just my assumptions and therefore what I would do are based on what’s important to me and what’s familiar to me. If I’ve got it wrong, then probably what I would do is not relevant!