My neice is five years old, and she is sick. She ...

I can only offer advice as it is found on website, but with the onset of vomiting, I would definitely seek medical attention at the nearest emergency room or with the child's regular physician.

Over-the-counter (OTC) cough syrups. In winter, nonprescription cough syrups practically fly off the drugstore shelves. But the American College of Chest Physicians strongly discourages the use of these medications because they're not effective at treating the underlying cause of cough due to colds. Some contain ingredients that may alleviate coughing, but the amounts are too small to do much good and may actually be harmful for children. In fact, the college has strongly recommended against using OTC cough syrups or cold medicines for any child younger than 14. Coughs associated with a cold usually last less than three weeks; if a cough lingers longer than that, see your doctor. In the meantime, try soothing your throat with warm lemon water and honey and humidifying the air in your house. Avoid giving honey to infants.

source:  http://www.revolutionhealth.com/conditions/cold-flu/common-cold/treatment/cold-remedies?s_kwcid=ContentNetwork|853883574 

Infant Cough/Cold Medicines Recalled

FDA reviewers have said the product labels "should include prominent language to describe the risk of overdose in children." They also recommended dropping the statement on many over-the-counter remedies urging parents to "consult a physician" about use of decongestants in children under 2 and antihistamines in children under 6.

Medicines that are being recalled from the U.S. market include: Concentrated Infants' TYLENOL Drops Plus Cold; Concentrated Infants' TYLENOL Drops Plus Cold & Cough; PEDIACARE Infant Drops Decongestant (PSE); PEDIACARE Infant Drops Decongestant & Cough (PSE); PEDIACARE Infant Dropper Decongestant (PE); PEDIACARE Infant Dropper Long-Acting Cough; PEDIACARE Infant Dropper Decongestant & Cough (PE) products; Dimetapp Decongestant Plus Cough Infant Drops; Demetapp Decongestant Infant Drops; Little Colds Decongestant Plus Cough; Little Colds Multi-Symptom Cold Formula; Robitussin Infant Cough DM Drops; Triaminic Infant & Toddler Thin Strips Decongestant and Triaminic Infant & Toddler Thin Strips Decongestant Plus Cough.

(Additional reporting by Kim Dixon and Lisa Richwine in Washington, D.C.)

source:http://www.reuters.com/article/newsOne/idUSN1133248720071011 

This question doesn't take any brain power at all. 

Take your niece to the doctor. If you can get into her normal doctor (i.e. quick appointment), that's the better way. If that can't be accomplished, take her to ER. You mention no other symptoms that might help in diagnosis, but coughing to that level alone is enough.  If she's got a fever at or over around 101, I'd have here in there instantly. 

With children and health issues, there is NO excuse to wait.

As Nike says "Just DO It." 

Take her to a doctor. Trying to treat her yourself can result in criminal prosecution (practicing medicine s/ a license.) Plus, it's what you need to do. Anything standing in the way of this?