Mix a cup of epsom salt in enough warm (slightly warmer than her bath) water to fill the bottom third or so of a pot big enough for her foot. Have her soak it for 30 minutes. See if you can see the shard (a blacklight, instead of normal indoor lighting, helps).
If nothing is visible, put neosporin on it and stay off it - then repeat the soak every 6 hours or so.
The only reason a doctor or hospital would be needed is for infection - and unless you broke a bottle of bacteria or your child is severely immunocompromised, it's not going to be today.
The body's immune system will push the shard back to the top - and the very salt water will pull fluid from the surrounding tissues, to reduce swelling and draw it out quicker.
S.