To solve these problems, I made two inner compartments, one for my laptop, and one for other stuff. The compartments can be attached to each other and to the backpack.
Step 1: Preparations
Step 2: Sewing the Compartments
First, cut out the pieces of fabric you need. I only used the place mats to make the front of the compartments more sturdy, because the back is attached to the back side of the back pack or to the sturdy side of another compartment. So for the front part of the compartment you'll need two identical pieces of fabric. Pin the two pieces together and sew the sides and the bottom. Don't sew the top yet, since you will have to put in the place mat after sewing the compartment together.
Next, sew the fabric for the sides together and pin them on the front of the compartment. Sew the sides to the front.
Before you sew the back side of the compartment to the sides, make four button holes in the corners (for attaching the compartment to the back side of the back pack later). Take into account the location of the screws of the back side of the backpack (which I didn't, so I had to leave out one screw). Then pin and sew the back side to the compartment.
Turn the compartment outside-out. You'll only need to put in the place mat and close the front side of the compartment. Unfortunately, you cannot do this before putting the compartment together, because you won't be able to turn the compartment outside-out when the place mat is already in. I closed the front side by hand.
Repeat this step for each additional compartment you want to make.
I made two compartments and made buttons to attach them. Stitch four buttons to the corners of the front side of the first compartment.
Step 3: Attaching the Compartment(s) to the Backpack
First, you cut four pieces of a bike tube. You stich the buttons on the pieces. Next, you determine where on the back side you'll need to glue the buttons. I used Bison Kit for this. Put glue on both the pieces of bike tube and on the back side of the back pack. Wait ten minutes and glue the buttons on the back side.
Attach the compartment to the back side of the back pack with the buttons. When you want to put the back side in the back pack again, remove the compartment first, otherwise you won't be able to attach the screws again.
Step 4: Extra Features
On the second compartment, I made pen holders on the inside, and a pocket for my cell phone on the outside.
Step 5: Result
It is best to use a fabric that is quite sturdy. The red compartment was made of an old sweat vest, which turned out to be a bit too infirm.