Ip – Setting subnet mask to 255.255.255.255

How can we calculate the subnet mask (255.x.x.x) for this given IP ?

You can't. To find the prefix, you need the address and mask, or the address and prefix length.

See the answers to this question on how to calculate everything.

There are a couple of things you could do:

• Change the network mask to expand the address range. For example, change the mask length to /23 to double , or /22 to quadruple, the number of network addresses. This is really less than ideal.
• Add VLANs. Each VLAN could have a separate network. If all your switches are layer-3 switches, you should be using those for LAN routing. You have not detailed your topology, but there are a few ways you could do this.

No matter what, you are going to need to change your security to take into account the new address range. The firewall is going to need to know all the new addressing so that it will not discard traffic to or from networks that it doesn't know. If it is performing NAT, it will need to know to NAT those source addresses, too.

If I make a switch my gateway (192.168.16.106), and set a static route on my firewall to send all 192.168.10.0/24 to that switch, then wouldn't the layer-3 switch just route the traffic for me?

Your proposed gateway (192.168.16.106) is not on the same network (192.168.10.0/24) as the hosts which will use it. Your host gateway needs to be set in the same network as the host is. The host gateway is used to send traffic destined to a different network because the host doesn't know how to get to the different network. How do you send traffic destined for a different network to a gateway on a different network if you don't know how to get to the different network in the first place?