I'm not a native English speaker so my grammar isn't perfect. I make some of mistakes especially when I'm speaking fast. I would like my child to have the cognitive advantage that comes from being bilingual so her father and I decided we would speak different languages to our child - I would speak in English, while he speaks in our mother tongue (an indian language which has a completely different grammatical structure to English). If I'm using incorrect grammar, will my daughter also learn incorrect English? Or will her brain automatically fix the mistakes as she is in the critical period of language development?
This depends. Your child can only learn what she gets from the input. So, if you consistently say 'goed' instead of 'went' or drop the third person singular endings on verbs ('she run'), that's what your daughter will learn to say.
However, no input is perfect (even from native speakers), so if you make some 'errors', your daughter is likely to make the correct generalizations anyway. But there are aspects of English usage/grammar - e.g. tenses or articles - where non-native input may result in acquiring non-standard usage patterns.
Of course, you are also unlikely to limit her input to just your speech. From very early on (18 months or so), it would make sense to start showing her English-language videos for children, so she should get sufficient input there, even if she's not exposed to other speakers in person.
But you should also be aware that this approach is not always successful. See this recent blog post looking at research on the topic: https://www.psychologytoday.com/blog/life-bilingual/201504/one-person-one-language-and-bilingual-children.
Fortunately for your family, kids will typically 'fix' inconsistent language input of their elders! The most famous case of this is a Deaf child named Simon, who almost exclusively learned ASL from his parents who were late learners of ASL themselves, and hence didn't quite have native command of the language. While his parents inconsistently used ASL morphology, Simon regularized this input, which is to say he correctly used the ASL morphology at levels equal to children exposed to native signers.