I'm sorry for your loss and welcome to the forum.
Did you check the sex of the new piggy yourself to confirm it is also female?
When bonding, it is done in a specific process. Putting them in side by side cages doesnt really start the process - it just allows them to know there is another piggy in the vicinity. The bonding doesnt really start until they are put into a neutral territory bonding pen for several hours.
They need to be put into a neutral territory bonding pen (somewhere your original piggy does not see as her own territory) together for several hours (overnight if needs be) so they can see if they like each other, accept each other and will be able to form a bond and hierarchy. On your lap isnt going to give then freedom of movement and the hours they need in the bonding pen to see if they will work together.
If they accept each other then you thoroughly clean down the cage they are to live in (so the original piggy who lived in the cage no longer sees it as her territory) and then after several hours in the bonding pen they can be moved into the cage to start their life together. You will see a lot of dominance behaviours during those hours in the bonding pen as well as the following two weeks after a successful bonding as it takes that long for them to establish a relationship.
Unfortunately, buying another piggy on spec does mean you run the risk that they will not like each other and will not bond. They have to be compatible in character to be able to form a relationship and if they dont like each other upon neutral territory introduction, then the bonding will fail and they must live in side by side cages permanently so they can only interact through the bars only.
Look at the guides I have added in below so you can gauge where things are between them to help you decide if things are normal bonding or if they are actually failing.
Nipping and a bit of hair pulling will happen but if they are both doing it aggressively and neither is going to back down, then that means they dont like each other and would need to be kept separately I'm afraid.
Bonding and Interaction: Illustrated social behaviours and bonding dynamics
Dominance Behaviours In Guinea Pigs
Sows: Behaviour and female health problems (including ovarian cysts)