ssd hard drive disappears after ca 30 seconds

OK, great! This is good news.
The diagnostics indicate that the drive is not dead, and your files would then be safe and available to retrieve off the drive.
You are almost there.

So here is the thing:
You cannot start into Windows using the problem drive. Windows was installed on your old PC and all configurations are set up for that. Windows will not start in your new PC with the old drive.

You can't find and move your files without getting into Windows.

You must use the drive that came with your current, working PC to start into Windows. That SSD drive must be connected before you can do anything else.

To retreive your files from the old SSD it MUST be connected as a second drive along with the new SSD drive.

Does your motherboard have additional SATA ports (connections)?
Do you have a second SATA data cable you can use?
Do you have a second SATA power connector you can use?

Once you get both SSD's connected and start the PC it is possible that the system will be confused about which Windows installation to boot from. For this reason you would want to use the F12 Boot Menu to select the correct SSD.

To make this easier you might want to check to see what name the new SSD is given in the Boot Menu before connecting the old SDD.

AnnaChino said:

Omg you have no idea how good that sounds!!!

Concerning the cables there is an additional cable with red, yellow, black etc wires already in the tower (that's the power cable, right?) I can attach the old ssd there, I think.

Yes it is. Yes you can.

That is the power connector you would use.
The White connector is a SATA connector. You can use that.

I wonder whether I could use the cable from my old computer?
Yes you can.

(I mean the other, smaller one that's flat and blue or black) Or is that a silly idea, because it's most likely been destroyed/blown?
That's a possibility, but if the drive is still good then the cable probably is too.
If you use it, and you can't see the drive after, then you should try a new cable.

If that's the case then I would have to get a new one, but that can easily be solved. Also, I don't know if I could use the white socket to connect the old ssd?
Yes you can. The color difference normally means that it is a slower SATA port. But that does not matter in this case because you are only trying to get your data.

I uploaded the image so you can take a look. If that's not a socket I can use, then...uhm... I'll think of something and find a computer with multiple sockets....

When I have both ssd's attached I would make sure in the boot menu that Dell chooses the new ssd to start from and then I can access the old ssd, is that correct?
Yes, that's correct. You would click to highlight the correct SSD and then hit ENTER.

Here is another option for you:
There are 2 SATA cables already attached to the motherboard in your picture. If one of those is connecting an optical drive (CD/DVD drive) then you could just disconnect it from the optical drive (leave it connected in the motherboard) and use that to connect the old SSD.
Obviously you won't be able to use the CD/DVD during the process, but you shouldn't need it to just get your files.

Congratulations on your persistence... though I do think you've been fortunate.

It's not just a question of backing up data- but also of preserving your whole O/S.

Routinely use disk imaging - endlessly recommended here... that creates a compressed copy of the used part of the partitions imaged as a file which you can save on external storage- which can then be stored safely, clear of theft, lightning, the dog etc..

E.g. Macrium Reflect /Aomei Backupper (free/paid)..

Then if your disk(s) fail you can restore things as was to a new disk from your disk image.
The image file can also be mounted and files accessed with file explorer - a full backup.

I additionally schedule a System Restore point to be created daily (tutorial) - and have addiitonal backup for selected key data htat changes more often.

Think what may go wrong- and know you have a way to recover from it, relatively quickly.

Fantastic! Good work.

I will add to dalchina's excellent advice that when you get some extra cash you might want to think about getting a good quality surge protector strip to plug your expensive components into. These are not all that expensive and add an extra layer of protection for power surges and lightening strikes.

Excellent help by TV2, ably supported by dalchina and persistent intake by the thread starter @annaChina in spite of many unfortunate events. Kudos to all three . ( @dalchina, I knew that your call was too premature to be taken. .)