Engine shut off while driving?

Hello, I bought the vehicle listed above diesel for my daughter yesterday. She has broken down twice and can't get it to start at all. It was juddering at a junction and stopped, no lights on and not turning over. She managed to get someone out to help who swapped the battery but still not started, it won't jump start either. The engine management light came on. It briefly got going and was in limp mode but has since died again. What are the most common problems it can be? It was a private sale, and I don't think I will have any comebacks. It was great for the test drive, no problems at all and getting it home. I have put a battery tester on, and the battery is flat but still won't go even after a charge. All help appreciated, thanks.