Thank you very much for your help! I got the result:
function Base.delete!(l::Node, index::Int64)
l_before = l
l_after = l
for i = 1:(index-2)
l_before = l_before.next
end
for i = 1:(index)
l_after = l_after.next
end
l_before.next = l_after
return l
end