I've been trying to install the expansion pack from the GOTY version disc 2 for a few hours now on my newer Xbox 360 Slim console, but while I have managed to install KoTN, Shivering Isles still refuses. Oblivion and Dashboard are version 1.2.1 and 2.0.12611.0 respectively. I have a Live silver account but I've noticed I get disconnected whenever the disc is launched (I'm using wi-fi).

Basically, what happens is that once launched, the disc asks about installing an update. I choose yes and it very swiftly installs and moves on to the actual DLC installation. Since I have KoTN working, I select only SI to install and wait until the install bar reaches about halfway before it stops the installation, saying that some content didn't get installed. Checking the memory, I see no SI there. I've also tried installing SI with both DLCs selected, but no cigar.

Any help, please?

You should first delete all oblivion content from hard drive, then clear the cache. Now You should start from Disc2 - install all content first on HDD then insert Oblivion Disc1 and install it (press Y on game icon)
BTW - do You have a HDD in console?? If You have only 4GB of internal memory You probably will have to get HDD.

By "Oblivion content", does it include the saves or just the installed content? And yes, I have a 250 GB HDD with more than enough space, so memory shouldn't be an issue.

Right then, so I transferred my latest saves to a USB, deleted everything about Oblivion from the HDD and cleared the system cache. I even disconnected from the wi-fi. I then launch the second disc, which asks me to install an update. I choose yes and it swiftly finishes that (and by swiftly I mean I didn't even see a progress bar). I go on to installing the expansion pack, noticing that the shivering isles is slightly more greyed out than KotN. I choose to install both, patiently waiting. The KotN installs fine, but SI stops in the middle and throws at me the message "Can't install one or more items." I'm not connected to Xbox Live, there's no instance of Oblivion of any form or shape on the Xbox 360 250 GB Slim and it still refuses to install. The disc has been used on an older white Xbox 360 model, but I don't think that should impact the installation from a disc.

OK i have given that mater some thought: If You are not connected to XBL, do as follows:
1. Delete >>ALL<< data from HDD (including gamer profile)
2. Perform a HDD Format.
3. Clear the cache of HDD (can be named as "maintenance")
4. Turn off console, unplug cables then remove HDD from it's slot.
5. Wait for about 10 min.
6. Put back HDD, plug all cables and turn on console.
7. Create offline gamer profile.
8. Now You can insert Disc2 and try to install it. If it succeed, then You can install Oblivion disc1.

If the problem still occur, it means that there is something wrong with HDD. Maybe if You have replaced that HDD from old 360 model, it can not work 100% properly on new 360 slim (maybe different data management on HDD)

PS: if Your 360 slim have the internal 4GB mermory, You MUST format it too and clear it's cache.
PS2: There is some possibility that You will have to buy original HDD for the new 360 slim model. There may be problem with compatibility between old "FAT" HDD and new "SLIM" HDD