I did the same at an ivy league school. I was informed I could go on a visitors visa (90 day) which is incorrect because I was clearly not going to visit. Being British, they said I didn't need to apply for anything if it was under 3 months but telling pokies at the border is a bad thing - especially in your case since they already have you on the system as being declined for a B1/2 - they already know your reasons for wanting to go there, so what will you say when you get there? They will just send you back.
Stop panicking. You need to contact the international office at the university and tell them you have been accepted for ..... and need to apply for a J1. They should be able to get you to send them your mum's bank statements etc to see that you can afford to stay there during your internship. They will need to know details of your internship too. Then they'll issue a DS2019 - it only takes a few days. You then need that DS2019 to apply for your J1 visa, which should be approved the day you go to the embassy if you have done everything properly.
Don't be scared to approach their international office, they do this all the time for overseas students and visiting students etc. Enjoy your time in the US.