Rich, with the greatest of respect - before heading off into visas and the like, I think you would be well advised to calm down a bit, look at the facts and see if you can deal with them.
You say you don't think there is any sexual element in her work but you say she works all through the night. Maybe things are different in Korea but I doubt many guys decide to go for a plain massage at 4am. If it turned out that sex is involved, you need to decide whether you can deal with that now and in the long term.
Secondly you say her ex wants her back and that there has been friction between them recently. Ask yourself, based on what you know not what she tells you - is that relationship over?
I can see this all going South for you, I really can.
If you sort everything out then subject to the usual conditions, there is no reason why you can't apply for a settlement visa with her. The kids on the other hand could be a different matter. If the children's father is Korean you need to find out what the law on taking children abroad permanently is in Korea. I can't be sure of this but I think the British Embassy will want some form of proof that the father gives his permission for the kids to go to live in the UK.
I remember around 30 years ago, my ex met an American bloke and wanted to marry him and take my son to the USA. I point blank refused to give my permission as I would almost certainly never have seen him again.
I'm guessing you are new to all this and know little about UK visas and the requirements. Not only are you likely to need the father's permission - are you aware of all the requirements you girlfriend will have to meet in order to get a visa? Your minimum income requirements to support her (more with 2 kids) - the requirement for her to have a certain standard of English when she applies - the huge cost etc. etc. I'd start doing some reading - plenty on this site if you look for it.
"I imagined she would come to UK for a month or two, we get married here and then she goes back to Korea to get kids."
Are you thinking she can just visit here and you can get married? If so, you cannot. If you want to marry in the UK, you need to apply for a fiance visa - not a recommended route as it involves an extra visa. That said, I've never heard of anyone applying for a fiance visa who is bring their kids too - not sure if that's possible.
I don't want to go much deeper into the visa situation yet as I think you have some thinking to do but before it escapes me - keep this in mind. It is far easier to make a visa application to bring your wife and kids to the UK than it is for just her and the kids later. There can be all sorts of problems bringing them separately as I'm sure other members who've done it will agree.
Anyway, you have lots of reading to do but I would look at the relationship first. From the brief description you've written is seems like one of those that's going to end in tears - only you know the true state of your relationship. Above all, slow down.