I have a family in Shoreditch that fits perfectly, and even a birth certificate for Eliza Mills. But I do not have proof that she is my Eliza.
There are no birth "certificates" in England until 1837. Is the parish register baptism that Debra found in 1828 St Leonard Shoreditch what you are referring to?
[This Eliza Mills was baptised 2 March 1826 - there is an additional notation in the heading "by whom the ceremony was performed - this appears to give dates of birth for all baptisms. In Eliza's case it is 2 Nov 1826.]
You say you have a family in Shoreditch which fits perfectly - what exactly do you have? Have you found them on the 1841 census? (I had a quick look but could not easily see them - but I am on my ipad so searching is more laborious, so decided to leave it for now ...)
Eliza's birth noted as being in Peckham on her daughter's marriage certificate, may have been due to her siblings being born there and she mistakenly thought she was also born there, or perhaps she was born in Peckham but baptised in Shoreditch after the family moved there.