Sailvi,
Thanks for the extra detail but something still doesn't add up in the information provided so far. Do you have any links to the reports published by the 3 authorities you mentioned? What is clear is that the crew did an excellent job in incredibly trying circumstances. The thing that I still don't understand though is how the a/c ended up where it did. If you look at the photos provided by the link at post 2 and compare what you can see to a satellite view of the field available here;
Airport Maps, Charts, Diagrams - Barajas Airport - LEMD - MAD - Airport Guide
and also using a Jeppy chart 10-9B it is fairly easy to make out that the jet came to rest off the southern edge of taxiway J at point J2-3 and pointing towards the southerly end of terminal 4. That is quite some distance from any runway. It must have been one hell of a ride between them leaving the runway and the a/c coming to rest there. Either that or they vacated the runway in a controlled manner but lost steering later, which seems more likely. You don't mention loss of the reserve brake and steering system in you post which may be the key.