"Continued" is wrong. The past participle used as an adjective is passive -- it says someone or something made a continuation on her, which is nonsense.
"Continuing to work" is at least grammatically correct. But in fact the present participle by itself has the continuous sense you want:
Meanwhile she spent four years working at the bank while studying for her law degree.
Here "while" has its original and full meaning "at the same time as". It perfectly indicates that for four years she juggled work and study.
I've left the word "meanwhile" at the beginning of the sentence under the assumption that just before this sentence, the text mentioned other things were happening in her life during the time she was working and studying: she was seeing William. But if that's not the case, and the text makes no such mention, strike out "meanwhile" as elegant but unnecessary.
She spent four years working at the bank while studying for her law degree.
The comma after "bank" is not necessary.