Non-existent instance in Game Maker

First thing. object_get_name needs object index, not instance index, like this:

var on = object_get_name(other.object_index); // or maybe (other.id).object_index, I don't have ability for check it now

Second. You must check what instance_find returned. Because instance may not exists.

var obj = instance_find(enemyname, enemyid);
if instance_exists(obj)
{
    speed = obj.speed;
    direction = obj.direction;
}

Third. instance_find needs object_index, not an object's name (which is a string) and index (which is not equal id).

I don't understand why you don't use just id of the instance.

var spd = other.speed;
var dir = other.direction;
var inst_id = other.id;

with instance_create(other.x-lengthdir_x(new_push_amount, pd), other.y-lengthdir_y(new_push_amount, pd), fire)
{
    speed = spd;
    direction = dir;
    enemyid = inst_id;
}

Or same, but less lines:

var obj = instance_create(other.x-lengthdir_x(new_push_amount, pd), other.y-lengthdir_y(new_push_amount, pd), fire);
obj.speed = other.speed;
obj.direction = other.direction;
obj.enemyid = other.id;

And Step End event (not Step, otherwise there will be a lag):

if instance_exists(enemyid)
{
    speed = enemyid.speed;
    direction = enemyid.direction;
}
else
{
    instance_destroy(); // ? I don't know, is need you it or not
    // but anyway don't forget to destroy object
}