Indeed Brenda you have quite a challenge ahead of you. As Jay said you may have to get off Twitter to draw people to it. How about email combined with an Instagram campaign, since Instagram is a tad more Twitter friendly than it's owner, Facebook. But there may be a way to use a Facebook campaign to draw people to Twitter using a hashtag.
One of the most obvious too is Twitter Ads. Can't hurt to have your client's profile appear amongst targeted potential followers. It would savy your tired fingers a lot of work.
Third, try partnering with a local business that will allow you to promote your Twitter campaign using their brick and mortar biz and their own Twitter profile. Since you're already hosting an event together, have a Twitter party.
Do something during the event that will make them pull out their phones and follow your client on the spot. Of course you gotta offer some freebie to make it enticing. Kind of like door prizes, request people follow at the event to win something then announce winner.
Fourth, try a campaign that's less about about "the get" and more about "the give". Partner with a non-profit. Say once you hit your goal you'll donate $1100 to a charity or donate time or service once the goal is made. This way the 11000 serves a bigger purpose than just getting followed.