When I got my private, I took four weeks vacation and got it done within that time frame. I had no flight experience at all before I started. Since you already have 10 hours logged, it should be doable. I scheduled two flight lessons each day, with a two hour break in between, during which I had lunch and hit the books. On days we couldn't fly because of the weather, we did ground.I did not go to a pilot mill, but picked a small FBO with four flight instructors. They knew I had a time limit and worked with me. I also picked a school far away from home, so I had no distractions, and nothing else to do other than study.
For me personally, that was the way to go. When I started with my instrument, I took one lesson a week. That didn't work for me at all. There was just too much time in between.