Hello & welcome, I am Dr. B, a licensed veterinarian and I would like to help you with your wee one today. I do apologize that your question was not answered before. Different experts come online at various times; I just came online, read about your wee one’s situation, and wanted to help.
Again, I do apologize that my colleagues were unable to assist you sooner. If Lily has eaten rat poison, then this is quite serious. Most of our rat bait products will cause harm to dogs just as it does with rats, and most work by causing problems with the dog's ability to clot their blood.
Since it has been a few hours, you are likely past the time frame where the induction of vomiting would be beneficial. Still there are means to limit absorption. If you haven't already had Lily seen by her vet, you will want to either do so right away (since this is an emergency) or consider administering activated charcoal (LINK) available from the pharmacy to bind any remaining material in her stomach. For activated charcoal, we tend to give 4 grams per pound is given every 8 hrs. This can be mixed with food or water to syringe feed (do note that it stains, so keep it away from white carpets/clothes). It works to bind the non-absorbed material and can limit the dose she has had.