Leaflet calculating meters per pixel at zoom level

You can use this to work out the metres per pixel:

metresPerPixel = 40075016.686 * Math.abs(Math.cos(map.getCenter().lat * Math.PI/180)) / Math.pow(2, map.getZoom()+8);

You can use the containerPointToLatLng conversion function of L.Map to get the latLngcoordinates for a given pixelcoordinate. If you take one of the first pixel, and one of the next, you can use the distanceTo utility method of L.LatLng to calculate the distance in meters between them. See the following code (assuming map is an instance of L.Map):

var centerLatLng = map.getCenter(); // get map center
var pointC = map.latLngToContainerPoint(centerLatLng); // convert to containerpoint (pixels)
var pointX = [pointC.x + 1, pointC.y]; // add one pixel to x
var pointY = [pointC.x, pointC.y + 1]; // add one pixel to y

// convert containerpoints to latlng's
var latLngC = map.containerPointToLatLng(pointC);
var latLngX = map.containerPointToLatLng(pointX);
var latLngY = map.containerPointToLatLng(pointY);

var distanceX = latLngC.distanceTo(latLngX); // calculate distance between c and x (latitude)
var distanceY = latLngC.distanceTo(latLngY); // calculate distance between c and y (longitude)

That should work, thanks to Jarek Piórkowski for pointing my mistake before the edit.