Welcome @At-end-of-the-tether to the Community. I hope you find your time here very worthwhile and continue to be an active part of the community.
The following is only general advice, I am not a legal expert and so cannot give you formal advice.
As it is a contract for next day in home service, they seem to be breaking the ACL (Australian Consumer Law) requirements. It should only take them a few minutes on the phone to determine if you have it plugged in correctly. After that your contract should mean they send someone out to do the troubleshooting and if needed arrange repair or replacement.
The fact it isn’t charging is probably beyond most standard users to determine why. Remind Dell of their requirements under ACL, in writing (this can be email), phone calls are next to useless when arguing Consumer Law as they are almost without proof of what was discussed. Give them a reasonable time to respond (in this case probably 24 to 48 hours) and state what you want as a remedy eg come and repair it within 24 hours as per the contract. If they don’t respond you are probably within your rights to take it elsewhere to be repaired and charge Dell for the cost. If it is beyond economical repair you could seek replacement or refund of the computer. You can also seek refund of your contract cost and compensation for any loss you have suffered.
CHOICE offers free templates for emails/letters as does the ACCC. CHOICE members can get some advice on how to proceed by using the CHOICE Help service. You can also get free legal advice from many places that will help you decide what to do.
If after writing and you don’t receive a satisfactory resolution you can then go to your Office of Fair Trading and/or your Civil and Administrative Tribunal. Having the proof of that written contact with Dell is important to allow you to complain to these bodies.
