You deployed Eunice too late, which means liches fired a free round to your Alyssa. That ruined the target priority of all your healers and your timings with deployments as well. That's why liches slowly started destroying your army, one by one. Losing Katie fairly early on is not an issue since you had enough UP to take on the liches. The problem with that map is that completing it with those units is like cooking a good dish: you have to be methodic and spot on with your decisions. One mistake, and your plan will be ruined as you saw.

3 mages can wipe the bulk of enemies that come before the liches. You have to focus on deployment and withdrawal to keep those pesky liches entertained with beefy units. Your Alyssa can take up to 2 shots, but not more than that. On the mages: we would need to know their affection aside from the level. Male units tend to have better base stats but less trust growth, while females have worse base stats but better affection growth. The reason why Eth uses 3 Valeries is to use their skill to quickly clean the clusterfuck at the end and focus on dealing with these liches as fast as possible.