you shouldnt delete a service becasue you dont use it ;), you should go into services and disable it if you belive that you dont use it!
have you tried iisreset? or restarting your server? i presume its a test server? also as important when you say you removed it, what did you do exactly to remove the service? are you talking about services.msc? or from a folder or from central admin? knowing what you did we can add it back in and solve the current issue.
also like to note that have you updated sharepoint server to october? or is it only at march update? if so you need to update it to the latest version, you need the march update to update to any other latest update!
http://technet.microsoft.com/en-us/sharepoint/jj891062.aspx
its indicating that your using a build version of 15.0.0.0 on assembly Microsoft.Office.TranslationServices but it should be at a higher build.
goto:
C:\Windows\Microsoft.NET\assembly\GAC_MSIL\Microsoft.Office.Web.Common\v4.0_15.0.0.0__71e9bce111e9429c\
now find the assembly Microsoft.Office.TranslationServices and right click, select properties and than check its build version! I think the build version is ment to be higher than just 15.0.0.0
a bit about the service you removed:
http://msdn.microsoft.com/en-us/library/microsoft.office.translationservices.aspx
lets see if you can get it running again:
run the following in powershell:
$tj = get-sptimerjob "Sharepoint Translation Services"
$tj.Runnow()
http://msdn.microsoft.com/en-us/library/jj163145
did you run SharePoint 2013 Products Configuration Wizard after the update? any update you make you need to run SharePoint 2013 Products Configuration Wizard.
if that powershell code fails than you need to run:
installutil assemblyname.exe
assemblyname.exe is the dll name of Microsoft.Office.TranslationServices located within C:\Windows\Microsoft.NET\assembly\GAC_MSIL\Microsoft.Office.Web.Common\v4.0_15.0.0.0__71e9bce111e9429c\
http://msdn.microsoft.com/en-us/library/50614e95(v=vs.110).aspx
when its successful you can then go into services and disable not delete ;).
to sort the original issue out you either run microsoft office repair or sharepoint 2013 repair
for sharepoint 2013 repair
Control Panel -> Programs and Features for SharePoint Server 2013
any issues run powershell command from here:
http://stevemannspath.blogspot.co.uk/2013/02/sharepoint-2013-repairing-installation.html
EDIT
based on the answers you have given me i would say that there is a conflict between central admin that is looking for dll version 15.0.0.0 and what you actually have on the file system which is 15.0.4505.1000 and obviously its failing when its looking for a method that doesnt exist in the new version which you have!
to try and get around this issue as you cant seem to add the service back in you could try going to central admin web.config:
C:\inetpub\wwwroot\wss\VirtualDirectories\
goto the above location and then go into the central admin folder (usual named from its port number which you can see in the url of central admin page)
once you have opend it up you can scroll down till you see the web.config file, open it with your selected text application like notepad or notepad++ and look for the dll Microsoft.Office.Web.Common.
once you have found the dll Microsoft.Office.Web.Common you can change the buil version from 15.0.0.0 to 15.0.4505.1000. now save and exit.
do an iisreset and try to add the service back in again! lets hope this simple trick works!
