My songs on my iPhone won't download onto my new laptops' iTunes?

I have all my songs on my iPhone and my (old) laptop with iTunes. My old laptop got stolen, and I got a new one. I didn't have iCloud on my old one. There's no way of getting the old laptop back and I can't transfer the songs onto the new one for some reason without buying them again. They're all on the same account, and if I do try and add the songs onto the new laptop, it says that I can only have 1 iTunes library per device.

I've tried syncing it, but it doesn't show up in my library.

When I do this it also asks me

"Are you sure you want to remove existing music, movies, TV shows, books, and tones from this iPhone and sync with this iTunes library?

Music, movies, TV shows, books, and tones synced to "My Phone iPhone" from other iTunes libraries will be removed and items will be synced from this iTunes library."

I remove and sync then when I press done this pops up:

"You have changed the settings for the iPhone "My Phone iPhone". Would you like to apply these changes?"

I click apply, then this pops up:

"The iPhone "~~My Phone~~ iPhone" is synced with another iTunes library. Do you want to erase this iPhone and sync with this iTunes library?

An iPhone can be synced with only one iTunes library at a time. Erasing and syncing replaces the contents of this iPhone with the contents of this iTunes library."

I don't want to erase the music off of my phone, because it's the only way I can get it onto the new laptop.So, what do I do?

Ok - this is what you do:

go into your itunes library which I'm assuming is the library you used on your old stolen laptop that is you are signing into the same account?

open itunes

at the top left you will see a small black and white square
click on the square
click show menu bar
click store
click on view my apple id

now look down the list and click on de authorize all
this means no one can use the stolen laptop on itunes because you have de authorized its use.
now go back into store and click on authorize this computer:

You have to make sure you are signing in to the same library that was on the stolen laptop

Do not use a new library:

If you do not see your purchases in the library this is how you get them back: You do not need to buy them again:

go into store and at the right you will see 'purchased'
click on purchased
click all songs

you will see a list of all your songs
go right to the bottom of the page and click on the cloud that says download all.

Now you have your purchases back:

When you sync your iphone just say yes you want to over right the old library with this one, if you don't over right you will be back to square one: Now you have to erase and sync and trust me it works: we have covered everything

I'm sorry this is so long but I'm trying to give you a step by step approach and I hope it helps:

If you still have any problems at all please edit at the bottom of your question, I will watch the question and I will edit you back in my answer so keep checking:

I hope my answer proves useful to you and would appreciate best answer if I have resolved your issue