We're about to send out our annual survey to our users - it's such an important thing to be able to gather all that information on sentiment and get a really good insight into what your users think about your community.
If you're worried about a low uptake and you know that $50 can fix this, then pay for it yourself and make sure that you get the budget allocated for next time.
We use SurveyMonkey - it's an excellent platform.
Think very carefully about what questions to ask the community - only ask questions that will be really useful for you and which you can act upon - get rid of anything that you already know or can find out the answer to - eg, you may know where the member is from and how old they are - or your analytics package may already give you the info you need.
I wouldn't worry about the length of the survey too much - just make sure that you set expectations up front in your invite - eg, 'If you have a moment, please could you fill in a short survey about the community. There are 18 questions and we estimate that it should take 10 minutes of your time. All completed surveys will be entered into a prize draw to win...'
The survey will tell you an awful lot about your community and I'd say that it is very relevant and useful work to be doing.
If you expect a big response then make sure that the questions can be evaluated by computer (eg. The questions should be multiple choice lists of options, Likert scales etc)
Have some open text answer fields but remember that someone will have to read and evaluate every one of these. A good example of an open question is, 'What single technical fix or extra function would improve the community experience for you?' You'll get loads of requests and ideas from something like that. And when you come to add new features, you'll be adding what most of the users have asked for.
Hope some of this helps,