Hi. Here is a quick rundown of the points I would keep in mind:
1. These are great schools and opportunities. I would say that Kellogg is significantly stronger for MBB recruiting and salaries both in the US and outside but your best ROI will be in the US of course. I hear from a number of folks recruiting for consulting and the comp they are offered in EU is 1/2 of what they would get in the US
2. Everyone recruits in London - INSEAD, HEC, LBS, Oxford, Cambridge, IESE, and anyone who speaks English so it is a pretty competitive space. I have a good friend who is graduating from LBS this year and he is still without an acceptable offer and not for the lack of trying though you never know how it will work in your case.
3. The most important part - relationships and MBA. Business school will put a tremendous pressure on your relationships. Most long-distance relationships will not survive it. Most close relationships will be stressed. Many reasons for it but ultimately you will be drifting apart unless you are together and since you are not married/serious status, there is a chance you break up (not as likely with marriage for example) or she kicks you out

so this is one of the sensitive and tricky aspects of it: 1) How serious are you and if they are the most important part of your life, then why are we even discussing this. If they are not, then Choose Kellogg 2) Depending on how involved your girlfriend is and how open you are, you should try to make the decisions with her input. Sometimes people get antsy and they are looking for a moment to break the news (e.g. I am not into you) and she may be having some thoughts. She may not be ready to share them quite yet but you need to get her buy in and that she is taking the responsibility for this decision as well since you will be in it together....
There is also a question where you want to be in the long run. If you want to be in Europe, then LBS is a great choice but if you would like to be in the US for a number of years, only Kellogg will get you there and if your GF wants to be there too, this will be the most likely opportunity to relocate in a few years once you get your H1B.
Forgive me if I am crude and insensitive. I feel the choice is in some sense between Kellogg and the relationship rather than Kellogg and LBS. There are people who have totally maintained relationships over the years in business school but it is very hard because you will be a changed person, your interests will be different, she will be a changed person and you will have little in common after 2 years of school, internships, etc. She would meet someone and you would meet suitable partners over these 2 years. I am sure your relationship is different but usually people will have a few fights, misunderstandings, etc. Tricky.