This season has finally come to an end with the Champions League finale. I really appreciate Inter's high effort in that game also. They did a really solid defending mostly and this caused Manchester City to have much bigger difficulties than normal. Maybe Inter couldn't win the Champions League title for the 4th time in their history but this was still a solid season for them.
Even though they couldn't win the Serie A title also they won 2 titles anyway - Italian Super Cup and Coppa Italia. Being that much competitive for the Champions League should also be considered as success for Inzaghi. Because Inter aren't a regular when it comes to Champions League finals.
Therefore I believe that Inter should keep Inzaghi at least for one more season. He has one more season to go on the contract already though.

Inter's struggle should be appreciated, even after losing they are still an extraordinary club. The reason is Manchester City, who are always able to score above 2-3 goals, but here Inter is showing that they are playing well. Able to withstand the aggressive City players. Scoring 1 goal means that Inter's defense is really solid. When compared to Madrid, who lost to Manchester City, the results were worse.
As a representative of the Italian club, of course Inter is a lucky club, even though in Serie A only finishing in 3rd position means that next season Inzaghi's troops will return to play in the UCL with much better performance.
For Inter Milan it was a general boost for their confidence and also for their regained recognition in the soccer world. I think they will benefit more from their Champions League season than they suffer from the defeat. Next year they can confidently play against the best teams in Europe and know that they have a chance. They have lost a bit of that confidence over the years and now they have all reason to believe in themselves again and actually aim higher than before.