You're definitely on the right track in terms of possible relationships (cross referencing the cM counts you gave with the ones http://thegeneticgenealogist.com/wp-content/uploads/2016/06/Shared-cM-Project-Version-2-UPDATED-1.pdf does indeed suggest a 2C / 2C1R or comparable relationship).
It sounds as if you understand everything correctly so far - it's just that there's no magic way of knowing the connection, as you've noticed!
To interrogate things further you'll need to look at both paper genealogy and DNA evidence. In particular, checking the X-DNA match could help to narrow things down - see below. I'd get as much info as you can from the cousin (e.g. her father's year of birth and location of birth if available).
Assuming you've built out your own family tree, you may find yourself in a position to try and speculatively place this person's father in your tree based on where and when he was born.
In terms of DNA evidence:
- If you can find other known 2nd cousins of your dad's and get them to test, this would be ideal. You could see who matches who, and narrow things down this way. In practice you obviously might not be able to find these people.
- On a basic level, Ancestry's shared matches shows profiles for others who have taken the test and match both parties at at least a certain number of cM (I believe 20cM in total). For matches this strong, this tool will almost certainly contain some leads for you to follow up.
- However, Ancestry's tools don't give you any access to the specific segments on which you match. If you're digging deeper, you'll definitely need this info. I'd suggest having the cousin upload his DNA to Gedmatch (http://www.gedmatch.com), and doing the same for your father if you haven't already (since your father shares more, and all the DNA you share is a subset, I'd focus on your father's DNA rather than yours).
- Gedmatch will generate 'kit numbers' for each uploaded DNA test result. With these kit numbers, you can use many powerful tools, including a list of all people who match both your father and the cousin. You can then view a report that lists out the segments in common for these people. If you can find people who are related to both your father and the cousin, and who share the same segments, this should help a great deal.
- The first thing I'd do is run an 'X-match' to see what X-DNA match (if any) your father has with the cousin. As she is female, she will have inherited X-DNA from her mother and from her father. Your father will only have inherited X-DNA from his mother. So if there is an X-DNA match between your father and the cousin, this would narrow the connection down to your father's maternal side.
- This also gives you a new pool of people to compare with (people upload from all the DNA test vendors as opposed to just Ancestry). If you're lucky, there will be some good matches at the top of your father's 'One to many matches' results page. With some work, you may be able to identify some relationships, which you can then use as further evidence.
Gedmatch can seem a bit impenetrable but is well worth looking into if you're sufficiently interested! There are some useful 'how to' videos (e.g. https://www.youtube.com/watch?feature=youtu.be&v=acGJmLlsWg4&app=desktop), and this is a good article: http://donnarutherford.com/?p=144