How do you calm down startled animals so they can be killed?

You can use the Kyne's Peace shout which,

will calm all wild animals, except frost trolls, within the range of approximately 100 paces.

or you can use one of the calming illusion spells Calm, Pacify, or Harmony which prevents animals or people up to a certain level from fighting. Also, as @WillihamTotland pointed out, Courage and Rally will make the animals run towards you instead of fleeing.

Another tactic you could try is using the paralyze spell or ice form shout to prevent them from running while you hack at them.

If you are simply not doing enough damage, there is the old trick of becoming a stealth archer, because sneaky archers do a lot of damage, and who hunts animals with daggers? All proper hunters use a bow.

This may be tedious, but you can read all of the stone tablets on the way up to High Hrothgar. This will add the Voice of the Sky blessing to your character, which lasts 24 real-time hours (20 in-game days):

Animals will neither attack nor flee from you.

There are ten stone tablets along the Seven Thousand Steps. They do not need to be read in any particular order.