Depends to a degree what was in the sick. Baby milk, strong coloured drinks, curry, alcohol.
First aid- use a spoon and from the outside scrape up as much as possible. Fold towel into a thick pile and stand on it so to absorb 'upwards' as much 'fluid' as possible.
Apply detol leave for 20 minutes and absorb with towel again. You may need to do this a few times.
As you only have home products I wouldn't do anymore than that. Sometimes this has worked but if not call in a pro if the carpet is worth it. If not go and hire a rug doctor.