hmm maybe the new computer u can`t open the case cos warranty, but the best way to do is, put ur old HDs as "slave" in ur new computer and it will acces the old HD normally then u transfer from D: to C: per exemple.
and by lan, u dont need 2 monitors for this.
u just need to share the entire HD on windows in ur old pc, then u just turn it on without monitor but with a lan cable pugled, then in ur new computer u just go in ur windows explorer and type \\ip(local) of ur old computer per exemple \\192.168.254.100
ull see the entire disk and ull have acess for this, just copy and paste ae. a tip ae, dont copy and paste all disk, go copying by folders and there u go my friend.