Work and carers allowance

DLA has stopped, you would have to apply for PIP, you won't get this if you are the diabetic unless you have other health issues.
If you are applying for a child, benefit would probably be at lower rate. You need to care for a person for no less than 35 hours per week. Having a job may affect any benefits.

You will only be entitled to Carers allowance if the person you care for receives one of the higher rates of PIP. If you are intending to work, there are some pretty complicated rules about earnings cut off points (if memory serves, its about £100 per week) and you would need to get expert advice on this.
Going back to PIP, if it's your daughter that you are claiming for, why would she need mobility, unless she has other issues? The care component yes, but I wouldn't have thought mobility would be necessary?

Children can qualify for Higher Rate Mobility component of DLA from age 3 years and Lower Rate Mobility from age 5 years. However, it's unlike that a child with type 1 diabetes would qualify for either rate of mobility, unless they have additional health issues.
For an adult to claim Carer's Allowance, they must provide at least 35 hours per week to a child in receipt of Middle or Higher Rate care component of DLA, or an adult who is in receipt of Standard or Enhanced Rate Daily Living component of PIP (Personal Indepence Payment). Plus to claim CA, the person must not be earning over a set amount per week.
Your best course of action is to consult a welfare rights organistation at your local council or Citizens Advice Bureau or similar body, depending on your local situation.