I feel sorry for your situation but it sounds like they sent the wrong person to the wrong place at the wrong time.
Speak to your superior to clear things up, maybe they have a different client/gig for you that suits your strengths and capabilities.
In regards to: "What can I do..":
Take management and leadership courses, get yourself a scrum certificate i.E., maybe train some interns or get a small project to get used to leading/managing.
"I'm quite introverted.." Reconsider if leading a team or managing a project is something you really want to do and you if are comfortable with it. There are lot of positions/roles in IT where other skills are required, and since you have 10 years of XP on your record you could find something different without too much hassle.
For a leadership-role you would need to find ways and learn how to become more extroverted, outspoken to voice your opinion, set disciplinary meassures, pat people on the back, speak to clients, make presentations and so on..