Hi MyGreenEyes,
After you have the sickle, just left click to kill the birds. If you die and get restored the birds that you've killed stayed killed so just keep on slicing and resurrecting and you'll get there.
Here's from another walkthrough --- The way to kill them is to wait until they actually dive at your head, and then click on them.
And another - Move slowly and click on the birds with the scythe when they are close to you.
MaG