You need to take this a bit more seriously. My youngest has been constipated for the last three years, but for the last year would literally go 10-14 days between movements.
Her colon is so stretched from the build-up that she can no longer sense when she has to go, so I cannot potty train her. She is devastated - I had to change her diaper in the nursery at church the other day (she just turned four today) and she actually said, "Mommy, then all the little kids will know that I pooped." She was SAD. Her doctors have advised me that she probably won't begin toilet training for BM's for almost a year, because that's how long it will take the damage in her colon to heal. She'll be in diapers until she is FIVE - a very bright little girl, who just physically can't learn to go when she needs to. THAT'S what you have to look forward to if you let this continue.
What breaks my heart is that we did everything we COULD to treat this, including daily Miralax, fiber supplements, lots of water, etc - and because of other health conditions, this just happened to her anyhow.
It's not something to joke about. Put her on Miralax, a full cap twice a day until she goes, and then continue one cap a day for a week, then half a cap - wean her back off of it. It won't likely make her go very quickly - it's not a laxative, it's a stool softener. It only has laxative properties in much larger amounts - like 2 capfuls at a time four times a day, which is the highest dose my daughter has ever had.
If she doesn't go within 24 hours, you can go get a bottle of Magnesium Citrate at the pharmacy for a dollar. An ounce of that stuff WILL make her go, believe me, but with THAT be prepared for some loose stools.
After this - if this is TRULY a battle of wills - you need to share with your daughter MY daughter's story. Ask her if she wants to go back to being in diapers and stay in them for the next year or two, and have to poop in a diaper and have the other kids know.