I totally agree with Woodpeckr.
If your problem is not solved by reinstalling the OS, and you already know that the slow boot was not caused by too many crapware running in the background when Windows starts, I would also try changing the HDD first, maybe also the RAM. It is not likely your motherboard is fried, otherwise your laptop would be totally dead.
http://www.irisvista.com/tech/laptops/Toshiba-Satellite-L555-L555D/disassemble-toshiba-laptop-1.htm
Read the instruction above. Your HDD and RAM are easily accessible by removing the covers of the corresponding compartments and pulling the components out (Steps 2 and 4). To remove the RAM, find two little tabs with bulby tips on each sides, pull them sideways / outward until the RAM "pops" up (you should see two little notches on either side of the RAM chip after puling it out - these are what the bulby tips catch onto and lock the RAM chip. When you replace the RAM, push it in firmly until you hear / feel it "click")
If your budget is $250, you can easily afford a new HDD plus new RAM.
Your HDD is the standard 2.5 inch SATA drive - it probably is a slower spinning (5400 rpm) one, and you should take the opportunity to upgrade to a faster (7200 rpm) standard drive with larger capacity, a solid state drive (SSD) (VERY FAST but more expensive per GB storage), or a hybrid drive (standard HDD for storage, combined with a small SSD for fast seek / swap). Some examples:
Seagate Momentus 750GB SATA2 7200RPM 2.5" LAPTOP HDD
OCZ Agility 4 128GB SATA3 6GB/s 2.5" Solid State Drive
Seagate Momentus XT 500GB 7200RPM 2.5" Solid State Hybrid Drive
NOTE: Your HDD probably has SATA2 (there are 3 versions), but the newest SATA3 is backward compatible with older versions.
To install a new internal drive (whether it is HDD, SSD etc.), you need to format the drive first before inserting it into the laptop. To do this, you need to buy a 2.5 inch USB external drive enclosure such as this one:
http://www.canadacomputers.com/product_info.php?cPath=14_202&item_id=021791
(If you are getting a large capacity HDD 500GB or over, make sure you check the specifications to see if the external drive enclosure is compatible with the large capacity drive).
After inserting the drive into the enclosure and using USB cable to hook it up to another Windows PC, click "Windows (Start)" icon in bottom left, and type "format disk" in the box. Click "create and format hard disk partitions". A box pops up to "initialize disk" (keep clicking OK to complete the process, you don't need to make any selection other than the default ones). Move your mouse over the disk which says "unallocated", right-click and choose "New simple volume". Keep pressing "Next" button and finally the "Finish" button (ie. everything default). Your new drive should now be recognized as a drive with a letter (eg. (F)). Now you can unhook the USB, remove the drive from the enclosure and put it into your laptop, before inserting the Windows installation CD.
If you have not done so, you should turn the laptop over and note down the Windows licence number, so that you can put it in during installation. There should not be any problem when you use the old licence number to register with Microsoft, because upgrading hard drive is a common thing (it will not work if you change the motherboard!).
I cannot find the specifications of your Toshiba model but I suspect your RAM has 1066MHz speed ("DDR3 1066" or "PC3-8500" - you should refer to the specification on your user manual, or open up the RAM cover to look at what is written on the piece(s) of RAM chip. I also do not know the maximum amount of RAM your laptop can take (refer to your manual) - it may be a good time to buy more RAM to maximize it: this is useful if you do gaming or run many intense programs at the same time. Remember to tell the sales person you want "
RAM for laptop" or "SODIMM", which is different from RAM for desktop PCs. There is no harm buying RAM which has faster speed (eg. use "DDR3 1333" RAM for a computer using "DDR3 1066" RAM in the specification), because faster RAM is backward compatible. However, it is not recommended to "mix" RAMs of different speed. If you use a slower RAM, your laptop will not be able to run at full speed.