Not sure if I got Remilia's last lines right, so I'd be happy if someone could check that over. Don't have the confidence to really tackle the other lines, sorry.
できる takes the action to be done as the subject, i.e. with が (and occasionally as the direct object, i.e. を, maybe?) and the doer as the indirect object, i.e. with に, so the original translation of 私にもできる as "I can do it too" was right. That's also Remilia speaking, not Flandre.
Ah...Uh, I mean, of course I can make it!! But can a beginner like Flan make it!?Sa, Sakuya...Well-Yay yay!Yes, quite right.I think some sweets thatEVEN A BEGINNERcan EASILY makeare suitable at first,YES?Well... since it's easy... probably...If one were to say what is the easiest,cookies,or maybe pudding...
In Japan, "pudding" usually means sweet custard pudding (creme caramel).Teach me! Teach me, sis!Yes, my Lady?Umm,well..um...Pudding!!! Pudding!!!Can I make it too!!?PUDDING!*glance*