How old is your DD?
Emily is 21 months and she goes to bed at 6.25pm and wakes between 6.15-6.45am.
She is always really tired and whiney about 10-10.30am but she will not sleep.I take Jessica to school at 12.15pm and on the walk back (15 mins in all) she sleeps and I put her in the cot until she wakes up at about 2.30pm.
You could try the morning sleep.It's not that long to wait until bedtime,only an hour difference between am and pm nap)
You may find,depending on her age,that she has a short nap in the morning and a longer one in the afternoon.
Give it a go and see how she is.
Let me know how it goes