Never mind, I found it. I was looking in the wrong place. \YOUR_ADMIN\includes\templates\tpl_all_packing_slips.php is missing two lines of code that were in the original version.

After line 134, you need to add

<td class="main"><b><?php echo 'Shipping Method' ?></b></td>
<td class="main"><?php echo $order->info['shipping_method'] ?></td>

So the full code at that point is

<tr>
<td class="main"><b><?php echo ENTRY_PAYMENT_METHOD; ?></b></td>
<td class="main"><?php echo $order->info['payment_method']; ?></td>
<td class="main"><b><?php echo 'Shipping Method' ?></b></td>
<td class="main"><?php echo $order->info['shipping_method'] ?></td>
</tr>