If I’m a legal resident in the US but my daughter has lived here illegally for a few years, how do I apply for her without her having to leave?

If your goal is to find a way to get her an immigration benefit within the U.S, then you may want to consider becoming a Naturalized citizen first and then petition for her as the minor unmarried child of a U.S Citizen. So long as you are able to show she entered the U.S legally at some point and not inadmissible , then the entire process can be completed within the U.S without the need for her to go outside.

You can apply for her and given that she is a minor, she is not yet accruing unlawful presence time.  However, when the time comes for her to get her visa (in about 3 years when the priority is current, since you are a permanent resident and therefore your petition on her behalf will be a preference category petition and not an immediate relative petition) she WILL still need to leave the US to consular process for the immigrant visa but she will not be barred from reentry since she will not have accrued any unlawful presence time.  If you want to try to avoid her having to leave the US at all, you will need to wait until you are a US citizen and either petition for her then as an immediate relative or convert the petition you file now as a permanent resident to that of an immediate relative once you are a US citizen.  At that time, as long as you can document that your daughter ENTERED the US lawfully but overstayed and as long as she is under the age of 21 at that time, she will be able to adjust status to permanent residence within the US.