This one is a can of worms: The problem is that the DB application is taking an incredibly long time to start (10 minutes). Your job is to find out why and there are MANY possible reasons. A little intuitive thinking can go along way here however. Do the two computers access any other remote applications or the internet to compare load times with? Try FTPing a file over the link to one of the computers,is it slow? If it is, the problem is network related, if not, the problem is with the client/server app and this needs to be investigated (also look at how the app interfaces/transmits data over the network). The bit of information provided makes me think that the sole purpose of thse two machines is to access this DB and this makes your job a little harder. Now you need to use "external" tools to gather information about the problem:
If you manage the switches/routers between the 2 PC's and the external DB, you can look at port statistics or use SNMP to determine the general health of the 256k link. This will only show you if the WAN link is the culprit however, if not, you have to look elsewhere for clues.
Your best bet is to use a laptop loaded with sniffer software such as Ethereal (free for download) and break the network between the PC's and the 256k link (easiest if done close to the PC) so you can capture the DB traffic in action. This is were an understanding of TCP/IP or IPX/SPX is required to determine what the behavior is. For example, I have found underperforming SQL servers by just watching the TCP/IP traffic at the remote PC having problems. When TCP sends a request that the server cant fulfill in a resonable amount of time (200ms I think), it will send the requesting PC an acknowledgement, that the packet was received and that signals the remote PC to be patient for the data. My customers PC would hang for 5 minutes at a time and it was all because the SQL server was taking its time to process the request. My seeing the acknowledgement packet told me immediately that the network delivered the packet and that the SQL server was to blame. In short, a sniffer can tell you volumes. I can't recommend them highly enough.
Honestly I have just scratched the surface here and my analysis is very general above and I left out detail that may change the results I spell out. I need to ask many more questions and get more data to really begin to help you find an answer but I hope this helps, let me know.
Brian