Can't boot to windows after deleted EFI boot partition(windows and ubuntu)

Question

I have Ubuntu 14.04, then I don't Know when but I upgrade it until 15.10. Then it's stuck to upgrade 16.10, but the trial result for upgrading is really bad. My Ubuntu can't start. And then(Because that time i need urgently) i re-install other Ubuntu 14.04(old but still LTS), and successfully saved. But accidentally deleted my boot EFI partition. So the windows boot is gone(i replaced it.. T.T ).

The problem is not far from :

Accidentally deleted efi partition

Can't boot after deleting EFI boot- and Windows-partition

Recover windows 10 EFI Boot from dual boot with ubuntu

But idk, how to check where the main problem of mine..

And i was try some other solution too..

1. Edit for add custom menu

   sudo gedit /etc/grub.d/40_custom
   
   menuentry "WINDOWS10 (on /dev/sda4)" --class windows --class os { 
   insmod part_msdos insmod ntfs set root='(/dev/sda,msdos1)' search --no-floppy --fs-uuid --set=root 9E1EA0371EA009FB drivemap -s (hd0) ${root} 
   chainloader +1 
   }
   

   stuck :
   cant know list msdos1,etc

2. Force for add boot load

   sudo grub-install --root-directory=/mnt --boot-directory=/mnt/boot /dev/sda4
   

   error :
   grub-install: error: cannot find EFI directory.

3. Default re-check

   sudo update-grub

   stuck :
   dummy : do nothing (check additional info in bottom)

4. Recheck UUID :

   check with Gparted and boot-repair

   still stuck, and say something that point cant with automatically repair, and i didn't understand that.

5. recheck with boot info script:

sda1: __________________________________________________________________________

File system:       vfat
Boot sector type:  FAT32
Boot sector info:  No errors found in the Boot Parameter Block.
Operating System:  
Boot files:        /efi/Boot/bootx64.efi /efi/ubuntu/grubx64.efi 
                   /efi/ubuntu/MokManager.efi /efi/ubuntu/shimx64.efi

sda3: __________________________________________________________________________

File system:       
Boot sector type:  -
Boot sector info: 
Mounting failed:   mount: unknown filesystem type ''

sda4: __________________________________________________________________________

File system:       ntfs
Boot sector type:  Windows Vista/7: NTFS
Boot sector info:  No errors found in the Boot Parameter Block.
Operating System:  
Boot files:        /bootmgr /Windows/System32/winload.exe

sda5: __________________________________________________________________________

File system:       swap
Boot sector type:  -
Boot sector info: 

sda6: __________________________________________________________________________

File system:       ext3
Boot sector type:  -
Boot sector info: 
Operating System:  Ubuntu 15.10
Boot files:        /boot/grub/grub.cfg /etc/fstab 
                   /boot/extlinux/extlinux.conf

sda7: __________________________________________________________________________

File system:       ntfs
Boot sector type:  Windows Vista/7: NTFS
Boot sector info:  No errors found in the Boot Parameter Block.
Operating System:  
Boot files:        

sda8: __________________________________________________________________________

File system:       ext4
Boot sector type:  -
Boot sector info: 
Operating System:  Ubuntu 14.04.5 LTS
Boot files:        /boot/grub/grub.cfg /etc/fstab

============================ Drive/Partition Info: =============================

Drive: sda _____________________________________________________________________

Disk /dev/sda: 500.1 GB, 500107862016 bytes 255 heads, 63 sectors/track, 60801 cylinders, total 976773168 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 4096 bytes

Partition Boot Start Sector End Sector # of Sectors Id System

/dev/sda1 1 976,773,167 976,773,167 ee GPT

GUID Partition Table detected.

Partition    Start Sector    End Sector  # of Sectors System
/dev/sda1           2,048     1,128,447     1,126,400 EFI System partition
/dev/sda3       1,128,448     1,161,215        32,768 Microsoft Reserved Partition (Windows)
/dev/sda4       1,161,216   219,013,119   217,851,904 Data partition (Windows/Linux)
/dev/sda5     219,013,120   239,493,119    20,480,000 Data partition (Windows/Linux)
/dev/sda6     239,493,120   435,382,271   195,889,152 Data partition (Windows/Linux)
/dev/sda7     468,869,120   976,773,119   507,904,000 Data partition (Windows/Linux)
/dev/sda8     435,382,272   468,869,119    33,486,848 Data partition (Linux)

"blkid" output: ________________________________________________________________

Device           UUID                                   TYPE       LABEL

/dev/sda1        BAEE-D94B                              vfat       MBR ESP
/dev/sda4        9E1EA0371EA009FB                       ntfs       System
/dev/sda5        f1ce1224-1c07-4b55-8d0e-819ec92966fa   swap       
/dev/sda6        f89ba19a-53e9-4bb8-b1f3-e98eb3df61db   ext3       
/dev/sda7        70A6363BA63601E0                       ntfs       data
/dev/sda8        e3da3fc1-3cfd-49e0-9883-ab4b9f1f56d5   ext4       

================================ Mount points: =================================

Device           Mount_Point              Type       Options

/dev/sda1        /boot/efi                vfat       (rw)
/dev/sda4        /media/reigin/System     fuseblk    (rw,nosuid,nodev,allow_other,default_permissions,blksize=4096)
/dev/sda6        /media/reigin/f89ba19a-53e9-4bb8-b1f3-e98eb3df61db ext3       (rw,nosuid,nodev,uhelper=udisks2)
/dev/sda7        /media/reigin/data       fuseblk    (rw,nosuid,nodev,allow_other,default_permissions,blksize=4096)
/dev/sda8        /                        ext4       (rw,errors=remount-ro)

additional info :

reigin@reigin-K43SA:~/Downloads$ sudo update-grub
Generating grub configuration file ...
Found linux image: /boot/vmlinuz-4.4.0-31-generic
Found initrd image: /boot/initrd.img-4.4.0-31-generic
Found Ubuntu 15.10 (15.10) on /dev/sda6
done

I am still learning how to use Ubuntu, gladly if anyone could help me.. thanks before..