The social media firm is trying to get Musk to buy it.
On Friday, Mr Musk said he was walking away from the deal.
He said that he didn't know the number of fake accounts on the platform.
The Delaware court has been asked to order the richest person in the world to complete the merger.
Having mounted a public spectacle to putTwitter in play, and having proposed and then signed a seller-friendly merger agreement, Mr. Musk apparently believes that he is free to change his mind, trash the company, disrupt its operations, and so on.
The lawsuit accuses Mr Musk of a long list of violations of the merger agreement that have cast a pall over the company.
Taylor said that the site wanted to hold Musk accountable.
In the past month, the share price of the company has fallen more than 8% as Mr Musk questioned the number of fake and fraudulent accounts on the platform.
Mr Musk said on Friday that he was pulling out of the deal due to a lack of information about the accounts.
He said that the dismissal of executives meant that it wasn't living up to its responsibilities.