Hi,

Can you let us know what these commands return?

sudo dpkg --configure -a
sudo apt-get install -f
cat /etc/apt/sources.list

sudo dpkg --configure -a

returned nothing

sudo apt-get install -f

returned:

Reading package lists... Done
Building dependency tree
Reading state information... Done
0 upgraded, 0 newly installed, 0 to remove and 1 not upgraded.

cat /etc/apt/sources.list

returned:

## Note, this file is written by cloud-init on first boot of an instance
## modifications made here will not survive a re-bundle.
## if you wish to make changes you can:
## a.) add 'apt_preserve_sources_list: true' to /etc/cloud/cloud.cfg
##     or do the same in user-data
## b.) add sources in /etc/apt/sources.list.d
## c.) make changes to template file /etc/cloud/templates/sources.list.tmpl
#

# See http://help.ubuntu.com/community/UpgradeNotes for how to upgrade to
# newer versions of the distribution.
deb http://mirrors.digitalocean.com/ubuntu trusty main
deb-src http://mirrors.digitalocean.com/ubuntu trusty main

## Major bug fix updates produced after the final release of the
## distribution.
deb http://mirrors.digitalocean.com/ubuntu trusty-updates main
deb-src http://mirrors.digitalocean.com/ubuntu trusty-updates main

## N.B. software from this repository is ENTIRELY UNSUPPORTED by the Ubuntu
## team. Also, please note that software in universe WILL NOT receive any
## review or updates from the Ubuntu security team.
deb http://mirrors.digitalocean.com/ubuntu trusty universe
deb-src http://mirrors.digitalocean.com/ubuntu trusty universe
deb http://mirrors.digitalocean.com/ubuntu trusty-updates universe
deb-src http://mirrors.digitalocean.com/ubuntu trusty-updates universe

## N.B. software from this repository is ENTIRELY UNSUPPORTED by the Ubuntu
## team, and may not be under a free licence. Please satisfy yourself as to
## your rights to use the software. Also, please note that software in
## multiverse WILL NOT receive any review or updates from the Ubuntu
## security team.
# deb http://mirrors.digitalocean.com/ubuntu trusty multiverse
# deb-src http://mirrors.digitalocean.com/ubuntu trusty multiverse
# deb http://mirrors.digitalocean.com/ubuntu trusty-updates multiverse
# deb-src http://mirrors.digitalocean.com/ubuntu trusty-updates multiverse

## Uncomment the following two lines to add software from the 'backports'
## repository.
## N.B. software from this repository may not have been tested as
## extensively as that contained in the main release, although it includes
## newer versions of some applications which may provide useful features.
## Also, please note that software in backports WILL NOT receive any review
## or updates from the Ubuntu security team.
# deb http://mirrors.digitalocean.com/ubuntu trusty-backports main restricted universe multiverse
# deb-src http://mirrors.digitalocean.com/ubuntu trusty-backports main restricted universe multiverse

## Uncomment the following two lines to add software from Canonical's
## 'partner' repository.
## This software is not part of Ubuntu, but is offered by Canonical and the
## respective vendors as a service to Ubuntu users.
# deb http://archive.canonical.com/ubuntu trusty partner
# deb-src http://archive.canonical.com/ubuntu trusty partner

deb http://security.ubuntu.com/ubuntu trusty-security main
deb-src http://security.ubuntu.com/ubuntu trusty-security main
deb http://security.ubuntu.com/ubuntu trusty-security universe
deb-src http://security.ubuntu.com/ubuntu trusty-security universe
# deb http://security.ubuntu.com/ubuntu trusty-security multiverse
# deb-src http://security.ubuntu.com/ubuntu trusty-security multiverse

Hi,

Can you try

sudo dpkg -i --force-overwrite /etc/logrotate.d/nginx

which will attempt to overwrite the specific file that failed.
You could also do --force-all instead

I got this error now:

dpkg-deb: error: `/etc/logrotate.d/nginx' is not a debian format archive
dpkg: error processing archive /etc/logrotate.d/nginx (--install):
 subprocess dpkg-deb --control returned error exit status 2
Errors were encountered while processing:
 /etc/logrotate.d/nginx

Ah, ok, sorry. Let's try
sudo dpkg -i --force-overwrite /var/cache/apt/archives/nginx_1.8.0-1~trusty_amd64.deb