Siobhan-Marie O'Connor, her GB team-mate, led at the halfway point, but Miley powered back in the breaststroke to lead with just 50m left, only to be overhauled by Hungary's Katinka Hosszu who touched just ahead of her in 2min 05.78sec.
Miley, who had won three bronze medals in the event at previous European Short Course events, took silver in 2:06.21, a new British record. Another Hungarian, Zsuzsanna Jakabos took bronze in 2:06.66, with O'Connor, from Stockport, fourth in a personal best 2:08.85.
Earlier in the day, Miley, competing at her first major event since the Olympics, won her heat in 2min 08.44sec and was the second-fastest qualifier. She also had a personal best in the heats of the 50m breaststroke with a time of 32.21sec but just missed out on qualifying.
Her fellow Scot, Craig Benson, swam a personal best in the 100m breastroke heat when he was third in 59.40 to reach the semi-finals, but he too missed out on the final after finishing fifth in 59.08.