1 month in and no difference!!

You're a decent weight for your height so as stated above, it's going to be slow. You would have a smaller calorie deficit and that means inaccuracies are going to harm you more than someone with a deficit of 1000 calories. Use your food scale and accurate entries. Adjust your expectations and eat enough to fuel your activity.

Also, are you confusing BMR with TDEE? Always eat above your BMR and under your TDEE to lose weight.

You aren't doing anything wrong. Simply take your time while losing weight. Don't expect it to happen instantly. I think most people get frustrated way too early, starve themselves, and then give up. Simply keep eating the same calorie MFP offers you for your target weight-loss.

Sometimes its just that those days you are retaining too much water, or you still have food in you etc. There are some days, I "gain" 4.5 lb's, even though I simply eat the same calories every day. Thats completely normal. Just keep on working out, which will increase your metabolic rate, and follow MFP calorie goals. (Don't forget to add your daily calories in to it, so you aren't starving yourself). Say 1500 calorie goal, 400 calories burned, eat 1900 calories that day.

And just in case you're not doing so, make sure you weigh your meals, and cook them yourself if you can. Definitely 100% accurate that way. Sometimes some companies (especially unknown brands, like a store bought sandwich) write less calories than it really is on the label, Casey Neistat had a good video about it.

I think MFP has a downside in that it encourages most people to eat too little. When you are already close to your goal weight, the goal is no longer the number on the scale, but converting some of the remaining body fat into muscle. I think you might want to start paying attention to macros more than just the calorie number. It's actually one of my pet peeves about this app and the community. What works for someone who is obese will need to be tweaked a lot to work for someone who has far less to lose. Cardio alone won't have the same effect anymore. You need to add strength training if you want to see a real change in your body composition.

I am a good deal heavier than you, but here is an example of why MFP calorie numbers are not great. When I plugged my goal into MFP, it told me to eat 1200 calories. Meanwhile my BMR is 1500!!! I had my numbers done by an actual professional (after losing 60lbs on 1200 which is a VERY RESTRICTIVE DIET!!!) and now I eat up to 1800 calories a day and I'm losing on schedule. I have also lowered my body fat percent meaning that while the numbers haven't moved like they did on the restrictive diet, I am not damaging my muscles by letting them waste away like I have in the past. I am getting stronger and smaller at the same time. If you pay attention only to the scale, you WILL lose weight, but if you pay attention to fitness and nutrition, you will have far more control of your goals and results.

Now it does seem that most people here are kind of against doing these things because it's a lot of work. I believe my body, health and life are worth the work. I'll put two hours into finding a show on netflix I want to watch, I should be able to put a few hours a week into my body.