Your attempt has two grammatical errors:
- "To bite on the neck" is 首を噛む.
- There's no need to use the ta-form for 噛まれる (because of the "relative tense" rule).
After fixing these, we have:
This is okay, but for better readability, you may want to change the word order:
Note that this sentence means she had no direct control over 彼 and 吸血鬼. She only "set up the situation" (by shutting a door, etc). And it does not state whether or not the vampire actually bit him. In other words, she only attempted. If you need to clearly express that the vampire actually bit him, you'll have to rephrase the sentence, for example: