She is very young to be going abroad alone even when feeling great.
I do think though, that there is a difference, often not acknowledged, between depression and grief/sadness. If many, or even most , people would have similar feeling under the circumstances, then that is not a mental health issue, it is a natural reaction. What your daughter is feeling seems reasonable, given what has happened. I can understand that she needs to process her grief and pain. I am concerned that she wants to speed up this process, which really does just take time to heal. Going away can mean that you simply have the same painful feelings, but in a different environment, which in itself can then be very disheartening.
Also in my experience personally of grief and trauma, and with family members with serious mental health issues, too much time alone to brood is not helpful. Some quiet time yes, when your batteries are low and and you don’t have the energy for others, but not days on end alone.
Has your daughter had any grief counselling ? I would also suggest the Freedom Programme, as a way of helping her find strength now after the abusive relationship.
All of this is also much more difficult at the moment due to Covid , or I would suggest that she does something more structured, that has time with people that builds on some skill or interest, combined with quieter time to potter about. Something more like a retreat, learning yoga or surfing, or anything that reconnects her with something she likes, or her body . I remember reading about a women’s surf retreat on Jersey, as I thought of taking my teenager, just to have some time together, there must be other things like this, depending on her interests.
If she is on antidepressants, then I would be even more cautious about her going somewhere alone, as I feel that they take the edge off fear, so may inhibit her ability to assess risk or danger. In older people this is less of a concern as they have a body of experience to guide them, but in someone so young this is a worry (even when robust actually, I got into dangerous situations as a teenager simply from naivety) .
A compromise that others have suggested is a city break together, where you split up in the day to explore, then come together for supper, or lunch, or to do a specific thing. This won’t give her the “time to think” that she wants, but it will be safer, and avoid the difficult night time alone.
I do feel that a very sad and vulnerable 17 year old girl, alone in a foreign city, is not a good idea.