How can I troubleshoot a Route 53 hosted zone?

About a week ago, I followed the AWS Route 53 instructions for creating a subdomain hosted zone to handle traffic for "request.i-codedesign.com".

The hosted zone provisioned these NS servers:

ns-1009.awsdns-62.net.
ns-1198.awsdns-21.org.
ns-1.awsdns-00.com.
ns-1807.awsdns-33.co.uk.

Also a week ago, I updated our DNS Service (Dotster) to add the above Name Server records:

For a while this worked well, traffic was getting to the intended server.

Within the past week, it stopped working. What steps can I take to troubleshoot this? I've tried nslookup request.i-codedesign.com ns-1.awsdns-00.com, for example, but it just returns four sets of "DNS request timed out. timeout was 2 seconds."

EDIT

When I try from home, the address resolves and I can see my web app.

From here, this is the nslookup result:

C:\jhfrench>nslookup
Default Server:  Wireless_Broadband_Router.home
Address:  192.168.1.1

> set debug
> request.i-codedesign.com
Server:  Wireless_Broadband_Router.home
Address:  192.168.1.1

------------
Got answer:
    HEADER:
        opcode = QUERY, id = 2, rcode = NXDOMAIN
        header flags:  response, auth. answer, want recursion, recursion avail.
        questions = 1,  answers = 0,  authority records = 1,  additional = 0

    QUESTIONS:
        request.i-codedesign.com.home, type = A, class = IN
    AUTHORITY RECORDS:
    ->  home
        ttl = 3600 (1 hour)
        primary name server = home
        responsible mail addr = (root)
        serial  = 10274
        refresh = 3600 (1 hour)
        retry   = 3600 (1 hour)
        expire  = 3600 (1 hour)
        default TTL = 3600 (1 hour)

------------
------------
Got answer:
    HEADER:
        opcode = QUERY, id = 3, rcode = NXDOMAIN
        header flags:  response, auth. answer, want recursion, recursion avail.
        questions = 1,  answers = 0,  authority records = 1,  additional = 0

    QUESTIONS:
        request.i-codedesign.com.home, type = AAAA, class = IN
    AUTHORITY RECORDS:
    ->  home
        ttl = 3600 (1 hour)
        primary name server = home
        responsible mail addr = (root)
        serial  = 10275
        refresh = 3600 (1 hour)
        retry   = 3600 (1 hour)
        expire  = 3600 (1 hour)
        default TTL = 3600 (1 hour)

------------
------------
Got answer:
    HEADER:
        opcode = QUERY, id = 4, rcode = NOERROR
        header flags:  response, want recursion, recursion avail.
        questions = 1,  answers = 1,  authority records = 0,  additional = 0

    QUESTIONS:
        request.i-codedesign.com, type = A, class = IN
    ANSWERS:
    ->  request.i-codedesign.com
        internet address = 54.225.172.91
        ttl = 300 (5 mins)

------------
Non-authoritative answer:
------------
Got answer:
    HEADER:
        opcode = QUERY, id = 5, rcode = NOERROR
        header flags:  response, want recursion, recursion avail.
        questions = 1,  answers = 0,  authority records = 1,  additional = 0

    QUESTIONS:
        request.i-codedesign.com, type = AAAA, class = IN
    AUTHORITY RECORDS:
    ->  request.i-codedesign.com
        ttl = 900 (15 mins)
        primary name server = ns-1828.awsdns-36.co.uk
        responsible mail addr = awsdns-hostmaster.amazon.com
        serial  = 1
        refresh = 7200 (2 hours)
        retry   = 900 (15 mins)
        expire  = 1209600 (14 days)
        default TTL = 86400 (1 day)

------------
Name:    request.i-codedesign.com
Address:  54.225.172.91

$  ~  dig request.i-codedesign.com ns

; <<>> DiG 9.8.3-P1 <<>> request.i-codedesign.com ns
;; global options: +cmd
;; Got answer:
;; ->>HEADER<<- opcode: QUERY, status: NXDOMAIN, id: 6399
;; flags: qr rd ra; QUERY: 1, ANSWER: 0, AUTHORITY: 1, ADDITIONAL: 0

;; QUESTION SECTION:
;request.i-codedesign.com.  IN  NS

;; AUTHORITY SECTION:
i-codedesign.com.   10798   IN  SOA ns3.webminders.com. root.i-codedesign.com. 1316218819 10800 1800 604800 300

;; Query time: 42 msec
;; SERVER: 192.168.3.1#53(192.168.3.1)
;; WHEN: Fri Jun 20 15:46:30 2014
;; MSG SIZE  rcvd: 98

There are no NS records for that zone, which is a problem. The query I issued should return the four nameservers you quoted in your answer.

In your i-codedesign.com zone, you need to create an NS record for the request subdomain.

Regarding your name server records posted, those are for i-codedesign.com. You still need to create NS records for the requests subdomain.