My step daughter's Father is now dead and she would like me to adopt her so that she can genuinely feel part of our family. I would also like this to happen but what is involved in the legal process and how long does this take.
Also is there a large cost involved?

You need to see a sloicitor/lega advisor, but as there is no need to get the biologicl afather to agree, it shouldn't take too long. My partner adopted 2 of his ex wife's children after their biological fathers were off the scene, 1 had left and 1 had died. It was much easier for him to adopt the one whose father had died, only took a couple of months.

Just go down to your local Family Court and ask what is necessary. As long as you have a copy of the bio father's death certificate, you should be able to do this without an atty.

ScottGem is right. Where I live, there isn't a family court and this kind of stuff is handled through the town and county. Call your local village, town, city, county (whichever applies in your area) clerk's office ahead of time and find out where you can pick up the forms and where you need to file them if their isn't a family court close by. They can give you all the info you need to be able to complete this properly. If they can't, they can direct you to the appropriate office that can. Shouldn't be a huge expense.

If the birth father is deceased, then it should be a very routine and inexpensive process. Simply have a lawyer draw up the adoption agreement, submit it to family court along with proof of the birth father's demise (i.e. death certificate) and it's signed, sealed and delivered.