I practice in the COP - deputyship for property and financial affairs is a routine application dealt with "on the papers" ie there is no hearing of any sort of the application is unopposed (which is the case here.) The forms can all be found online and an application can be made online via a portal (recommended for P&A applications as much quicker than via paper/post.)
You can if you are confident with the forms do this yourself, except for the COP3 capacity assessment which a doctor, social worker or other suitable professional eg OT needs to complete. (GPs charge a fee as it's non NHS work and LA social workers usually don't complete them for non-LA applications. You can engage an independent SW privately to complete a capacity assessment if necessary.)
Have a look at the COP1A, COP3 & COP4, COP14PADep & COP15PADep forms to see what's involved. You don't need the COP1 form if applying online but reading it will give you an idea of what information you need to provide about P (the person to whom the application relates.) The gov.uk website contains a useful explanation about the process and the requirements to notify the person who lacks capacity and 3 others.
If you decide to do it yourself I'd recommend starting with obtaining the Capacity assessment. Also note the need to notify people no more than 3 months before you apply.
If you prefer to use a solicitor their fees are payable from Ps funds and I'd expect the only upfront payment you may be asked to make is for a capacity assessment with the legal fees paid after the deputy order is made and you have access to Ps finances.
Deputyship for Health & Welfare is much less "routine" and there is a tendency these days to regard appointment of a HW Deputy as exceptional rather than routine.
As PP have said if you do wish to pursue it is absolutely the case that prior discussion with professionals involved in care/social services etc is the best approach.
The main obstacle you face is that in theory it's unnecessary to appoint a HW Deputy as a "decision maker" will, if P lacks capacity to make a welfare decision, apply the Mental Capacity Act correctly, consult with all those interested in Ps welfare and make a best interests decision having carefully weighed up all the options and considered the benefits and burdens of each option from Ps perspective.
Back in the real world professionals often do not do that (to put it mildly!)
In a HW application it is the applicant's means that are considered when looking at fee exemption and remission.
My own recent experience of making HW deputy applications for clients is that they are a lengthy business, not always successful (despite the applicants being exemplary family carers) and you are probably best looking at information available via Mencap and other charities to see what outcomes others in a similar position to you have achieved before deciding whether to pursue this. If you do decide on a HW application it's entirely separate from the PA deputyship and the application is paper based and submitted via post - again worth looking at the gov.uk website for information.
Good luck