Copying Logins To A New Server

I have succeeded in reattaching the db to the new server, but the old logins do not work, as has been experienced by many people.

What follows was is a solution from an earlier post (last year) on this topic. I tried running the queries from Master in on the new server, and got the following error:
Server: Msg 2601, Level 14, State 3, Line 1
Cannot insert duplicate key row in object 'sysusers' with unique index 'sysusers'.
The statement has been terminated.

The sql queries I am using are:

insert sysxlogins(srvid, sid, xstatus, xdate1, xdate2, name, password, dbid, language)select srvid, sid, xstatus, xdate1, xdate2, name, password, dbid, language
from "oldServerName".master.dbo.sysxlogins
where name not in ('DISTRIBUTOR_ADMIN', 'REPL_PUBLISHER', 'REPL_SUBSCRIBER', 'TRACEUSER', 'SA')

AND

insert sysusers(uid, status, name, sid, roles, createdate, updatedate, altuid, password)
select uid, status, name, sid, roles, createdate, updatedate, altuid, password
from "oldServerName".master.dbo.sysusers
where name not in ('public', 'INFORMATION_SCHEMA', 'db_owner', 'db_accessadmin', 'db_securityadmin', 'db_ddladmin', 'db_backupoperator', 'db_datareader','db_datawriter', 'db_denydatareader', 'db_denydatawriter', 'guest', 'dbo')

The old server is set up as a linked server on the new server.

This suggests that the information is all in the new server. Any tips? TIA D. Lewis