I guess it's more of a context that's getting you perplexed.
二浪 = 二年浪人
"浪人(ろうにん)" is a person who could not pass an entrance exam to a school (usually university), and is still studying to get into the school. "浪人" status will be lifted once they are accepted to a university. Sine entrance exam is conducted only once a year in Japan (as far as I know at this point), "浪人" is a status given for a year. If you are not accepted to any university (or decide no to go to any) after a year of "浪人" status, then it means you repeat the same status twice, meaning "二年浪人". Japanese love to shorten words, so we shorten the "二年浪人" to "二浪". There are also "三浪", "四浪" ...etc. This "浪人" phenomenon generally happens as many Japanese hold the beliefs that so-called "elite university" will guarantee their career, as well as a higher socio-economic status.
Now this is up to "二浪". Next, "二浪中の身で" needs to be broken into two parts: "二浪中" which means "in the middle of the status of 二浪", and the other "の身で" which is an expression to describe your status, usually with a connotation of degradation compared to others. So this will be translated into "under/in the status of '二浪'".
"てしまい" is a run-off of a sentence. The sentence is supposed to continue after this clause if it's formal writing, but in casual style, we sometimes write or say this way to emphasize the event that happened up to the point, and to make listeners or readers anticipate what story comes next. So here, we don't know what happened after s/he dropped off the prep school after two years of failing to proceed to university/high school etc.
I hope this will help.