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I am a Solicitor and will assist you.
Please may I ask - is the house solely owned and in your name? has she ever paid the mortgage? a deposit? or renovated?
The reality is that your ex has no right to occupy your property.
Whilst she may have a very limited claim in respect of adding value to the property - this doesnt give her a right to occupy and what you have offered is more than sufficient to reimburse her for her contribution.
You need to give her a date to vacate by. If she doesn't go - then you could change the locks so she can't gain access.
Let me know if I can help you further.
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There is no set timescales - because she is not a tenant. You can choose what you consider as reasonable. one or two months would be very reasonable.
There is no change in the legal position because of your girlfriends depression.