I'm not sure if this can be called a fix but it's currently working and here's what I did to get it working -
1. Disable automatic driver update.
(My Computer > Properties > Advanced System Settings > Hardware > Device Installation Settings)
I selected "Never install driver software from Windows Update". I did this because Windows Update seemed to keep installing the wrong card - Radeon 8500 or something)
2. Uninstall ATI drivers and software (Program Features)
Might only be "ATI Catalyst...."
3. Uninstall drivers for both video cards (Device Manager > Uninstall)
Be sure to check the box for "Delete the driver software for this device"
4. Download the latest BETA ATI driver.
For me, this was version "13.25.18.1" - http://www2.ati.com/drivers/firepro/performance/13.25.18.1-beta-mobile-165762-hp.exe
5. Download the latest Intel 4400 driver. (listed as Intel® Iris™ and HD graphics Driver for Windows 7/8/8.1* 64)
For me, this was version "15.33.8.64.3345" - https://downloadcenter.intel.com/Detail_Desc.aspx?agr=Y&DwnldID=23405&ProdId=3719&lang=eng
6. Extract the drivers to separate folders. I downloaded the .zip files instead of exe's so I could install nothing but the drivers.
7. Update the ATI card driver first.
Device Manager > Basic Display Adapter (something like that), mine was showing Error Code 31) > Driver > Update Driver > Browse for drivers on my computer > (location of extracted Intel driver)
The ATI card should no longer show an error in device manager.
8. Next, update the Intel card driver.
Device Manager > Basic Display Adapter (the only display adapter left), > Driver > Update Driver > Browse for drivers on my computer > (location of extracted ATI driver)
The Intel card should no longer show an error in device manager.
9. Enable Deep sleep again in bios (if you disabled it)
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After this I tested the sleep by clicking Power > Sleep. Then hitting the power button and then powering it back on.
Please post back if this fixes your problem. It'll help me determine if this is really a solution.